Category Archives: calculus

Rules for Inequalities

If a, b and c are real numbers, then

  1. a < b \Longrightarrow a + c< b + c
  2. a < b \Longrightarrow a - c < b - c
  3. a < b \hspace{0.1in} and \hspace{0.1in}c > 0 \Longrightarrow ac < bc
  4. a < b \hspace{0.1in} and \hspace{0.1in}c < 0 \Longrightarrow bc < ac special case: a < b \Longrightarrow -b < -a
  5. a > 0 \Longrightarrow \frac{1}{a} > 0
  6. If a and b are both positive or both negative, then a < b \Longrightarrow \frac{1}{b} < \frac{1}{a}.

Remarks:

Notice the rules for multiplying an inequality by a number: Multiplying by a positive number preserves the inequality; multiplying by a negative number reverses the inequality. Also, reciprocation reverses the inequality for numbers of the same sign.

Regards,

Nalin Pithwa.

Set Theory, Relations, Functions Preliminaries: II

Relations:

Concept of Order:

Let us say that we create a “table” of two columns in which the first column is the name of the father, and the second column is name of the child. So, it can have entries like (Yogesh, Meera), (Yogesh, Gopal), (Kishor, Nalin), (Kishor, Yogesh), (Kishor, Darshna) etc. It is quite obvious that “first” is the “father”, then “second” is the child. We see that there is a “natural concept of order” in human “relations”. There is one more, slightly crazy, example of “importance of order” in real-life. It is presented below (and some times also appears in basic computer science text as rise and shine algorithm) —-

Rise and Shine algorithm: 

When we get up from sleep in the morning, we brush our teeth, finish our morning ablutions; next, we remove our pyjamas and shirt and then (secondly) enter the shower; there is a natural order here; first we cannot enter the shower, and secondly we do not remove the pyjamas and shirt after entering the shower. 🙂

Ordered Pair: Definition and explanation:

A pair (a,b) of numbers, such that the order, in which the numbers appear is important, is called an ordered pair. In general, ordered pairs (a,b) and (b,a) are different. In ordered pair (a,b), ‘a’ is called first component and ‘b’ is called second component.

Two ordered pairs (a,b) and (c,d) are equal, if and only if a=c and b=d. Also, (a,b)=(b,a) if and only if a=b.

Example 1: Find x and y when (x+3,2)=(4,y-3).

Solution 1: Equating the first components and then equating the second components, we have:

x+3=4 and 2=y-3

x=1 and y=5

Cartesian products of two sets:

Let A and B be two non-empty sets then the cartesian product of A and B is denoted by A x B (read it as “A cross B”),and is defined as the set of all ordered pairs (a,b) such that a \in A, b \in B.

Thus, A \times B = \{ (a,b): a \in A, b \in B\}

e.g., if A = \{ 1,2\} and B = \{ a,b,c\}, tnen A \times B = \{ (1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}.

If A = \phi or B=\phi, we define A \times B = \phi.

Number of elements of a cartesian product:

By the following basic counting principle: If a task A can be done in m ways, and a task B can be done in n ways, then the tasks A (first) and task B (later) can be done in mn ways.

So, the cardinality of A x B is given by: n(A \times B)= n(A) \times n(B).

So, in general if a cartesian product of p finite sets, viz, A_{1}, A_{2}, A_{3}, \ldots, A_{p} is given by n(A_{1} \times A_{2} \times A_{3} \ldots A_{p}) = n(A_{1}) \times n(A_{2}) \times \ldots \times n(A_{p})

Definitions of relations, arrow diagrams (or pictorial representation), domain, co-domain, and range of a relation:

Consider the following statements:

i) Sunil is a friend of Anil.

ii) 8 is greater than 4.

iii) 5 is a square root of 25.

Here, we can say that Sunil is related to Anil by the relation ‘is a friend of’; 8 and 4 are related by the relation ‘is greater than’; similarly, in the third statement, the relation is ‘is a square root of’.

The word relation implies an association of two objects according to some property which they possess. Now, let us some mathematical aspects of relation;

Definition:

A and B are two non-empty sets then any subset of A \times B is called relation from A to B, and is denoted by capital letters P, Q and R. If R is a relation and (x,y) \in R then it is denoted by xRy.

y is called image of x under R and x is called pre-image of y under R.

Let A=\{ 1,2,3,4,5\} and B=\{ 1,4,5\}.

Let R be a relation such that (x,y) \in R implies x < y. We list the elements of R.

Solution: Here A = \{ 1,2,3,4,5\} and B=\{ 1,4,5\} so that R = \{ (1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)\} Note this is the relation R from A to B, that is, it is a subset of A x B.

Check: Is a relation R^{'} from B to A defined by x<y, with x \in B and y \in A — is this relation R^{'} *same* as R from A to B? Ans: Let us list all the elements of R^{‘} explicitly: R^{'} = \{ (1,2),(1,3),(1,4),(1,5),(4,5)\}. Well, we can surely compare the two sets R and R^{'} — the elements “look” different certainly. Even if they “look” same in terms of numbers, the two sets R and R^{'} are fundamentally different because they have different domains and co-domains.

Definition : Domain of a relation R: The set of all the first components of the ordered pairs in a relation R is called the domain of relation R. That is, if R \subseteq A \times B, then domain (R) is \{ a: (a,b) \in R\}.

Definition: Range: The set of all second components of all ordered pairs in a relation R is called the range of the relation. That is, if R \subseteq A \times B, then range (R) = \{ b: (a,b) \in R\}.

Definition: Codomain: If R is a relation from A to B, then set B is called co-domain of the relation R. Note: Range is a subset of co-domain.

Type of Relations:

One-one relation: A relation R from a set A to B is said to be one-one if every element of A has at most one image in B and distinct elements in A have distinct images in B. For example, let A = \{ 1,2,3,4\}, and let B=\{ 2,3,4,5,6,7\} and let R_{1}= \{ (1,3),(2,4),(3,5)\} Then R_{1} is a one-one relation. Here, domain of R_{1}= \{ 1,2,3\} and range of R_{1} is \{ 3,4,5\}.

Many-one relation: A relation R from A to B is called a many-one relation if two or more than two elements in the domain A are associated with a single (unique) element in co-domain B. For example, let R_{2}=\{ (1,4),(3,7),(4,4)\}. Then, R_{2} is many-one relation from A to B. (please draw arrow diagram). Note also that domain of R_{1}=\{ 1,3,4\} and range of R_{1}=\{ 4,7\}.

Into Relation: A relation R from A to B is said to be into relation if there exists at least one element in B, which has no pre-image in A. Let A=\{ -2,-1,0,1,2,3\} and B=\{ 0,1,2,3,4\}. Consider the relation R_{1}=\{ (-2,4),(-1,1),(0,0),(1,1),(2,4) \}. So, clearly range is \{ 0,1,4\} and range \subseteq B. Thus, R_{3} is a relation from A INTO B.

Onto Relation: A relation R from A to B is said to be ONTO relation if every element of B is the image of some element of A. For example: let set A= \{ -3,-2,-1,1,3,4\} and set B= \{ 1,4,9\}. Let R_{4}=\{ (-3,9),(-2,4), (-1,1), (1,1),(3,9)\}. So, clearly range of R_{4}= \{ 1,4,9\}. Range of R_{4} is co-domain of B. Thus, R_{4} is a relation from A ONTO B.

Binary Relation on a set A:

Let A be a non-empty set then every subset of A \times A is a binary relation on set A.

Illustrative Examples:

E.g.1: Let A = \{ 1,2,3\} and let A \times A = \{ (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}. Now, if we have a set R = \{ (1,2),(2,2),(3,1),(3,2)\} then we observe that R \subseteq A \times A, and hence, R is a binary relation on A.

E.g.2: Let N be the set of natural numbers and R = \{ (a,b) : a, b \in N and 2a+b=10\}. Since R \subseteq N \times N, R is a binary relation on N. Clearly, R = \{ (1,8),(2,6),(3,4),(4,2)\}. Also, for the sake of completeness, we state here the following: Domain of R is \{ 1,2,3,4\} and Range of R is \{ 2,4,6,8\}, codomain of R is N.

Note: (i) Since the null set is considered to be a subset of any set X, so also here, \phi \subset A \times A, and hence, \phi is a relation on any set A, and is called the empty or void relation on A. (ii) Since A \times A \subset A \times A, we say that A \subset A is a relation on A called the universal relation on A. 

Note: Let the cardinality of a (finite) set A be n(A)=p and that of another set B be n(B)=q, then the cardinality of the cartesian product n(A \times B)=pq. So, the number of possible subsets of A \times B is 2^{pq} which includes the empty set.

Types of relations:

Let A be a non-empty set. Then, a relation R on A is said to be: (i) Reflexive: if (a,a) \in R for all a \in A, that is, aRa for all a \in A. (ii) Symmetric: If (a,b) \in R \Longrightarrow (b,a) \in R for all a,b \in R (iii) Transitive: If (a,b) \in R, and (b,c) \in R, then so also (a,c) \in R.

Equivalence Relation: 

A (binary) relation on a set A is said to be an equivalence relation if it is reflexive, symmetric and transitive. An equivalence appears in many many areas of math. An equivalence measures “equality up to a property”. For example, in number theory, a congruence modulo is an equivalence relation; in Euclidean geometry, congruence and similarity are equivalence relations.

Also, we mention (without proof) that an equivalence relation on a set partitions the set in to mutually disjoint exhaustive subsets. 

Illustrative examples continued:

E.g. Let R be an equivalence relation on \mathbb{Q} defined by R = \{ (a,b): a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}. Prove that R is an equivalence relation.

Proof: Given that R = \{ (a,b) : a, b \in \mathbb{Q}, (a-b) \in \mathbb{Z}\}. (i) Let a \in \mathbb{Q} then a-a=0 \in \mathbb{Z}, hence, (a,a) \in R, so relation R is reflexive. (ii) Now, note that (a,b) \in R \Longrightarrow (a-b) \in \mathbb{Z}, that is, (a-b) is an integer \Longrightarrow -(b-a) \in \mathbb{Z} \Longrightarrow (b-a) \in \mathbb{Z} \Longrightarrow (b,a) \in R. That is, we have proved (a,b) \in R \Longrightarrow (b,a) \in R and so relation R is symmetric also. (iii) Now, let (a,b) \in R, and (b,c) \in R, which in turn implies that (a-b) \in \mathbb{Z} and (b-c) \in \mathbb{Z} so it \Longrightarrow (a-b)+(b-c)=a-c \in \mathbb{Z} (as integers are closed under addition) which in turn \Longrightarrow (a,c) \in R. Thus, (a,b) \in R and (b,c) \in R implies (a,c) \in R also, Hence, given relation R is transitive also. Hence, R is also an equivalence relation on \mathbb{Q}.

Illustrative examples continued:

E.g.: If (x+1,y-2) = (3,4), find the values of x and y.

Solution: By definition of an ordered pair, corresponding components are equal. Hence, we get the following two equations: x+1=3 and y-2=4 so the solution is x=2,y=6.

E.g.: If A = (1,2), list the set A \times A.

Solution: A \times A = \{ (1,1),(1,2),(2,1),(2,2)\}

E.g.: If A = \{1,3,5 \} and B=\{ 2,3\}, find A \times B, and B \times A, check if cartesian product is a commutative operation, that is, check if A \times B = B \times A.

Solution: A \times B = \{ (1,2),(1,3),(3,2),(3,3),(5,2),(5,3)\} whereas B \times A = \{ (2,1),(2,3),(2,5),(3,1),(3,3),(3,5)\} so since A \times B \neq B \times A so cartesian product is not a commutative set operation.

E.g.: If two sets A and B are such that their cartesian product is A \times B = \{ (3,2),(3,4),(5,2),(5,4)\}, find the sets A and B.

Solution: Using the definition of cartesian product of two sets, we know that set A contains as elements all the first components and set B contains as elements all the second components. So, we get A = \{ 3,5\} and B = \{ 2,4\}.

E.g.: A and B are two sets given in such a way that A \times B contains 6 elements. If three elements of A \times B are (1,3),(2,5),(3,3), find its remaining elements.

Solution: We can first observe that 6 = 3 \times 2 = 2 \times 3 so that A can contain 2 or 3 elements; B can contain 3 or 2 elements. Using definition of cartesian product of two sets, we get that A= \{ 1,2,3\} and \{ 3,5\} and so we have found the sets A and B completely.

E.g.: Express the set \{ (x,y) : x^{2}+y^{2}=25, x, y \in \mathbb{W}\} as a set of ordered pairs.

Solution: We have x^{2}+y^{2}=25 and so

x=0, y=5 \Longrightarrow x^{2}+y^{2}=0+25=25

x=3, y=4 \Longrightarrow x^{2}+y^{2}=9+16=25

x=4, y=3 \Longrightarrow x^{2}+y^{2}=16+9=25

x=5, y=0 \Longrightarrow x^{2}+y^{2}=25+0=25

Hence, the given set is \{ (0,5),(3,4),(4,3),(5,0)\}

E.g.: Let A = \{ 1,2,3\} and B = \{ 2,4,6\}. Show that R = \{ (1,2),(1,4),(3,2),(3,4)\} is a relation from A to B. Find the domain, co-domain and range.

Solution: Here, A \times B = \{ (1,2),(1,4),(1,6),(2,2),(2,4),(2,6),(3,2),(3,4),(3,6)\}. Clearly, R \subseteq A \times B. So R is a relation from A to B. The domain of R is the set of first components of R (which belong to set A, by definition of cartesian product and ordered pair)  and the codomain is set B. So, Domain (R) = \{ 1,3\} and co-domain of R is set B itself; and Range of R is \{ 2,4\}.

E.g.: Let A = \{ 1,2,3,4,5\} and B = \{ 1,4,5\}. Let R be a relation from A to B such that (x,y) \in R if x<y. List all the elements of R. Find the domain, codomain and range of R. (as homework quiz, draw its arrow diagram);

Solution: Let A = \{ 1,2,3,4,5\} and B = \{ 1,4,5\}. So, we get R as (1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5). domain(R) = \{ 1,2,3,4\}, codomain(R) = B, and range(R) = \{ 4,5\}.

E.g. Let A = \{ 1,2,3,4,5,6\}. Define a binary relation on A such that R = \{ (x,y) : y=x+1\}. Find the domain, codomain and range of R.

Solution: By definition, R \subseteq A \times A. Here, we get R = \{ (1,2),(2,3),(3,4),(4,5),(5,6)\}. So we get domain (R) = \{ 1,2,3,4,5\}, codomain(R) =A, range(R) = \{ 2,3,4,5,6\}

Tutorial problems:

  1. If (x-1,y+4)=(1,2), find the values of x and y.
  2. If (x + \frac{1}{3}, \frac{y}{2}-1)=(\frac{1}{2} , \frac{3}{2} )
  3. If A=\{ a,b,c\} and B = \{ x,y\}. Find out the following: A \times A, B \times B, A \times B and B \times A.
  4. If P = \{ 1,2,3\} and Q = \{ 4\}, find the sets P \times P, Q \times Q, P \times Q, and Q \times P.
  5. Let A=\{ 1,2,3,4\} and \{ 4,5,6\} and C = \{ 5,6\}. Find A \times (B \bigcap C), A \times (B \bigcup C), (A \times B) \bigcap (A \times C), A \times (B \bigcup C), and (A \times B) \bigcup (A \times C).
  6. Express \{ (x,y) : x^{2}+y^{2}=100 , x, y \in \mathbf{W}\} as a set of ordered pairs.
  7. Write the domain and range of the following relations: (i) \{ (a,b): a \in \mathbf{N}, a < 6, b=4\} (ii) \{ (a,b): a,b \in \mathbf{N}, a+b=12\} (iii) \{ (2,4),(2,5),(2,6),(2,7)\}
  8. Let A=\{ 6,8\} and B=\{ 1,3,5\}. Let R = \{ (a,b): a \in A, b \in B, a+b \hspace{0.1in} is \hspace{0.1in} an \hspace{0.1in} even \hspace{0.1in} number\}. Show that R is an empty relation from A to B.
  9. Write the following relations in the Roster form and hence, find the domain and range: (i) R_{1}= \{ (a,a^{2}) : a \hspace{0.1in} is \hspace{0.1in} prime \hspace{0.1in} less \hspace{0.1in} than \hspace{0.1in} 15\} (ii) R_{2} = \{ (a, \frac{1}{a}) : 0 < a \leq 5, a \in N\}
  10. Write the following relations as sets of ordered pairs: (i) \{ (x,y) : y=3x, x \in \{1,2,3 \}, y \in \{ 3,6,9,12\}\} (ii) \{ (x,y) : y>x+1, x=1,2, y=2,4,6\} (iii) \{ (x,y) : x+y =3, x, y \in \{ 0,1,2,3\}\}

More later,

Nalin Pithwa

 

 

 

 

 

 

 

 

Set Theory, Relations, Functions Preliminaries: I

In these days of conflict between ancient and modern studies there must surely be something to be said of a study which did not begin with Pythagoras and will not end with Einstein. — G H Hardy (On Set Theory)

In every day life, we generally talk about group or collection of objects. Surely, you must have used the words such as team, bouquet, bunch, flock, family for collection of different objects.

It is very important to determine whether a given object belongs to a given collection or not. Consider the following conditions:

i) Successful persons in your city.

ii) Happy people in your town.

iii) Clever students in your class.

iv) Days in a week.

v) First five natural numbers.

Perhaps, you have already studied in earlier grade(s) —- can you state which of the above mentioned collections are sets? Why? Check whether your answers are as follows:

First three collections are not examples of sets but last two collections represent sets. This is because in first three collections, we are not sure of the objects. The terms ‘successful persons’, ‘happy people’, ‘clever students’ are all relative terms. Here, the objects are not well-defined. In the last two collections, we can determine the objects clearly (meaning, uniquely, or without ambiguity). Thus, we can say that the objects are well-defined.

So what can be the definition of a set ? Here it goes:

A collection of well-defined objects is called a set. (If we continue to “think deep” about this definition, we are led to the famous paradox, which Bertrand Russell had discovered: Let C be a collection of all sets such which are not elements of themselves. If C is allowed to be a set, a contradiction arises when one inquires whether or not C is an element of itself. Now plainly, there is something suspicious about the idea of a set being an element of itself, and we shall take this as evidence that the qualification “well-defined” needs to be taken seriously. Bertrand Russell re-stated this famous paradox in a very interesting way: In the town of Seville lives a barber who shaves everyone who does not shave himself. Does the barber shave himself?…)

The objects in a set are called elements or members of that set.

We denote sets by capital letters : A, B, C etc. The elements of a set are represented by small letters : a, b, c, d, e, f ….etc. If x is an element of a set A, we write x \in A. And, we read it as “x belongs to A.” If x is not an element of a set A, we write x \not\in A, and read as ‘x does not belong to A.’e.g., 1 is a “whole” number but not a “natural” number.

Hence, 0 \in W, where W is the set of whole numbers and 0 \not\in N, where N is a set of natural numbers.

There are two methods of representing a set:

a) Roster or Tabular Method or List Method (b) Set-Builder or Ruler Method

a) Roster or Tabular or List Method:

Let A be the set of all prime numbers less than 20. Can you enumerate all the elements of the set A? Are they as follows?

A=\{ 2,3,5,7,11,15,17,19\}

Can you describe the roster method? We can describe it as follows:

In the Roster method, we list all the elements of the set within braces \{, \} and separate the elements by commas.

In the following examples, state the sets using Roster method:

i) B is the set of all days in a week

ii) C is the set of all consonants in English alphabets.

iii) D is the set of first ten natural numbers.

2) Set-Builder Method:

Let P be the set of first five multiples of 10. Using Roster Method, you must have written the set as follows:

P = \{ 10, 20, 30, 40, 50\}

Question: What is the common property possessed by all the elements of the set P?

Answer: All the elements are multiples of 10.

Question: How many such elements are in the set?

Answer: There are 5 elements in the set.

Thus, the set P can be described using this common property. In such a case, we say that set-builder method is used to describe the set. So, to summarize:

In the set-builder method, we describe the elements of the set by specifying the property which determines the elements of the set uniquely.

Thus, we can write : P = \{ x: x =10n, n \in N, n \leq 5\}

In the following examples, state the sets using set-builder method:

i) Y is the set of all months of a year

ii) M is the set of all natural numbers

iii) B is the set of perfect squares of natural numbers.

Also, if elements of a set are repeated, they are written once only; while listing the elements of a set, the order in which the elements are listed is immaterial. (but this situation changes when we consider sets from the view-point of permutations and combinations. Just be alert in set-theoretic questions.)

Subset: A set A is said to be a subset of a set B if each element of set A is an element of set B. Symbolically, A \subseteq B.

Superset: If A \subset B, then B is called the superset of set A. Symbolically: B \supset A

Proper Subset: A non empty set A is said to be a proper subset of the set B, if and only if all elements of set A are in set B, and at least one element of B is not in A. That is, if A \subseteq B, but A \neq B then A is called a proper subset of B and we write A \subset B.

Note: the notations of subset and proper subset differ from author to author, text to text or mathematician to mathematician. These notations are not universal conventions in math.

Intervals: 

  1. Open Interval : given a < b, a, b \in R, we say a<x<b is an open interval in \Re^{1}.
  2. Closed Interval : given a \leq x \leq b = [a,b]
  3. Half-open, half-closed: a <x \leq b = (a,b], or a \leq x <b=[a,b)
  4. The set of all real numbers greater than or equal to a : x \geq a =[a, \infty)
  5. The set of all real numbers less than or equal to a is (-\infty, a] = x \leq a

Types of Sets:

  1. Empty Set: A set containing no element is called the empty set or the null set and is denoted by the symbol \phi or \{ \} or void set. e.g., A= \{ x: x \in N, 1<x<2\}
  2. Singleton Set: A set containing only one element is called a singleton set. Example : (i) Let A be a set of all integers which are neither positive nor negative. Then, A = \{ 0\} and example (ii) Let B be a set of capital of India. Then B= \{ Delhi\}

We will define the following sets later (after we giving a working definition of a function): finite set, countable set, infinite set, uncountable set.

3. Equal sets: Two sets are said to be equal if they contain the same elements, that is, if A \subseteq B and B \subseteq A. For example: Let X be the set of letters in the word ‘ABBA’ and Y be the set of letters in the word ‘BABA’. Then, X= \{ A,B\} and Y= \{ B,A\}. Thus, the sets X=Y are equal sets and we denote it by X=Y.

How to prove that two sets are equal?

Let us say we are given the task to prove that A=B, where A and B are non-empty sets. The following are the steps of the proof : (i) TPT: A \subset B, that is, choose any arbitrary element x \in A and show that also x \in B holds true. (ii) TPT: B \subset A, that is, choose any arbitrary element y \in B, and show that also y \in A. (Note: after we learn types of functions, we will see that a fundamental way to prove two sets (finite) are equal is to show/find a bijection between the two sets).

PS: Note that two sets are equal if and only if they contain the same number of elements, and the same elements. (irrespective of order of elements; once again, the order condition is changed for permutation sets; just be alert what type of set theoretic question you are dealing with and if order is important in that set. At least, for our introduction here, order of elements of a set is not important).

PS: Digress: How to prove that in general, x=y? The standard way is similar to above approach: (i) TPT: x < y (ii) TPT: y < x. Both (i) and (ii) together imply that x=y.

4. Equivalent sets: Two finite sets A and B are said to be equivalent if n(A)=n(B). Equal sets are always equivalent but equivalent sets need not be equal. For example, let A= \{ 1,2,3 \} and B = \{ 4,5,6\}. Then, n(A) = n(B), so A and B are equivalent. Clearly, A \neq B. Thus, A and B are equivalent but not equal.

5. Universal Set: If in a particular discussion all sets under consideration are subsets of a set, say U, then U is called the universal set for that discussion. You know that the set of natural numbers the set of integers are subsets of set of real numbers R. Thus, for this discussion is a universal set. In general, universal set is denoted by or X.

6. Venn Diagram: The pictorial representation of a set is called Venn diagram. Generally, a closed geometrical figures are used to represent the set, like a circle, triangle or a rectangle which are known as Venn diagrams and are named after the English logician John Venn.

In Venn diagram the elements of the sets are shown in their respective figures.

Now, we have these “abstract toys or abstract building-blocks”, how can we get new such “abstract buildings” using these “abstract building blocks”. What I mean is that we know that if we are a set of numbers like 1,2,3, …, we know how to get “new numbers” out of these by “adding”, subtracting”, “multiplying” or “dividing” the given “building blocks like 1, 2…”. So, also what we want to do now is “operations on sets” so that we create new, more interesting or perhaps, more “useful” sets out of given sets. We define the following operations on sets:

  1. Complement of a set: If A is a subset of the universal set U then the set of all elements in U which are not in A is called the complement of the set A and is denoted by A^{'} or A^{c} or \overline{A} Some properties of complements: (i) {A^{'}}^{'}=A (ii) \phi^{'}=U, where U is universal set (iii) U^{'}= \phi
  2. Union of Sets: If A and B are two sets then union of set A and set B is the set of all elements which are in set A or set B or both set A and set B. (this is the INCLUSIVE OR in digital logic) and the symbol is : $latex A \bigcup B
  3. Intersection of sets: If A and B are two sets, then the intersection of set A and set B is the set of all elements which are both in A and B. The symbol is A \bigcap B.
  4. Disjoint Sets: Let there be two sets A and B such that A \bigcap B=\phi. We say that the sets A and B are disjoint, meaning that they do not have any elements in common. It is possible that there are more than two sets A_{1}, A_{2}, \ldots A_{n} such that when we take any two distinct sets A_{i} and A_{j} (so that i \neq j, then A_{i}\bigcap A_{j}= \phi. We call such sets pairwise mutually disjoint. Also, in case if such a collection of sets also has the property that \bigcup_{i=1}^{i=n}A_{i}=U, where U is the Universal Set in the given context, We then say that this collection of sets forms a partition of the Universal Set.
  5. Difference of Sets: Let us say that given a universal set U and two other sets A and B, B-A denotes the set of elements in B which are not in A; if you notice, this is almost same as A^{'}=U-A.
  6. Symmetric Difference of Sets: Suppose again that we are two given sets A and B, and a Universal Set U, by symmetric difference of A and B, we mean (A-B)\bigcup (B-A). The symbol is A \triangle B. Try to visualize this (and describe it) using a Venn Diagram. You will like it very much. Remark : The designation “symmetric difference” for the set A \triangle B is not too apt, since A \triangle B has much in common with the sum A \bigcup B. In fact, in A \bigcup B the statements “x belongs to A” and “x belongs to B” are joined by the conjunction “or” used in the “either …or …or both…” sense, while in A \triangle B the same two statements are joined by “or” used in the ordinary “either…or….” sense (as in “to be or not to be”). In other words, x belongs to A \bigcup B if and only if x belongs to either A or B or both, while x belongs to A \triangle B if and only if x belongs to either A or B but not both. The set A \triangle B can be regarded as a kind of a “modulo-two-sum” of the sets A and B, that is, a sum of the sets A and B in which elements are dropped if they are counted twice (once in A and once in B).

Let us now present some (easily provable/verifiable) properties of sets:

  1. A \bigcup B = B \bigcup A (union of sets is commutative)
  2. (A \bigcup B) \bigcup C = A \bigcup (B \bigcup C) (union of sets is associative)
  3. A \bigcup \phi=A
  4. A \bigcup A = A
  5. A \bigcup A^{'}=U where U is universal set
  6. If A \subseteq B, then A \bigcup B=B
  7. U \bigcup A=U
  8. A \subseteq (A \bigcup B) and also B \subseteq (A \bigcup B)

Similarly, some easily verifiable properties of set intersection are:

  1. A \bigcap B = B \bigcap A (set intersection is commutative)
  2. (A \bigcap B) \bigcap C = A \bigcap (B \bigcap C) (set intersection is associative)
  3. A \bigcap \phi = \phi \bigcap A= \phi (this matches intuition: there is nothing common in between a non empty set and an empty set :-))
  4. A \bigcap A =A (Idempotent law): this definition carries over to square matrices: if a square matrix is such that A^{2}=A, then A is called an Idempotent matrix.
  5. A \bigcap A^{'}=\phi (this matches intuition: there is nothing in common between a set and another set which does not contain any element of it (the former set))
  6. If A \subseteq B, then A \bigcap B =A
  7. U \bigcap A=A, where U is universal set
  8. (A \bigcap B) \subseteq A and (A \bigcap B) \subseteq B
  9. i: A \bigcap (B \bigcap )C = (A \bigcap B)\bigcup (A \bigcap C) (intersection distributes over union) ; (9ii) A \bigcup (B \bigcap C)=(A \bigcup B) \bigcap (A \bigcup C) (union distributes over intersection). These are the two famous distributive laws.

The famous De Morgan’s Laws for two sets are as follows: (it can be easily verified by Venn Diagram):

For any two sets A and B, the following holds:

i) (A \bigcup B)^{'}=A^{'}\bigcap B^{'}. In words, it can be captured beautifully: the complement of union is intersection of complements.

ii) (A \bigcap B)^{'}=A^{'} \bigcup B^{'}. In words, it can be captured beautifully: the complement of intersection is union of complements.

Cardinality of a set: (Finite Set) : (Again, we will define the term ‘finite set’ rigorously later) The cardinality of a set is the number of distinct elements contained in a finite set A and we will denote it as n(A).

Inclusion Exclusion Principle:

For two sets A and B, given a universal set U: n(A \bigcup B) = n(A) + n(B) - n(A \bigcap B).

For three sets A, B and C, given a universal set U: n(A \bigcup B \bigcup C)=n(A) + n(B) + n(C) -n(A \bigcap B) -n(B \bigcap C) -n(C \bigcup A) + n(A \bigcap B \bigcap C).

Homework Quiz: Verify the above using Venn Diagrams. 

Power Set of a Set:

Let us consider a set A (given a Universal Set U). Then, the power set of A is the set consisting of all possible subsets of set A. (Note that an empty is also a subset of A and that set A is a subset of A itself). It can be easily seen (using basic definition of combinations) that if n(A)=p, then n(power set A) = 2^{p}. Symbol: P(A).

Homework Tutorial I:

  1. Describe the following sets in Roster form: (i) \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} word \hspace{0.1in}  PULCHRITUDE\} (II) \{ x: x \hspace{0.1in } is \hspace{0.1in} an \hspace{0.1in} integer \hspace{0.1in} with \hspace{0.1in} \frac{-1}{2} < x < \frac{1}{2} \} (iii) \{x: x=2n, n \in N\}
  2. Describe the following sets in Set Builder form: (i) \{ 0\} (ii) \{ 0, \pm 1, \pm 2, \pm 3\} (iii) \{ \}
  3. If A= \{ x: 6x^{2}+x-15=0\} and B= \{ x: 2x^{2}-5x-3=0\}, and x: 2x^{2}-x-3=0, then find (i) A \bigcup B \bigcup C (ii) A \bigcap B \bigcap C
  4. If A, B, C are the sets of the letters in the words, ‘college’, ‘marriage’, and ‘luggage’ respectively, then verify that \{ A-(B \bigcup C)\}= \{ (A-B) \bigcap (A-C)\}
  5. If A= \{ 1,2,3,4\}, B= \{ 3,4,5, 6\}, C= \{ 4,5,6,7,8\} and universal set X= \{ 1,2,3,4,5,6,7,8,9,10\}, then verify the following:

5i) A\bigcup (B \bigcap C) = (A\bigcup B) \bigcap (A \bigcup C)

5ii) A \bigcap (B \bigcup C)= (A \bigcap B) \bigcup (A \bigcap C)

5iii) A= (A \bigcap B)\bigcup (A \bigcap B^{'})

5iv) B=(A \bigcap B)\bigcup (A^{'} \bigcap B)

5v) n(A \bigcup B)= n(A)+n(B)-n(A \bigcap B)

6. If A and B are subsets of the universal set is X, n(X)=50, n(A)=35, n(B)=20, n(A^{'} \bigcap B^{'})=5, find (i) n(A \bigcup B) (ii) n(A \bigcap B) (iii) n(A^{'} \bigcap B) (iv) n(A \bigcap B^{'})

7. In a class of 200 students who appeared certain examinations, 35 students failed in MHTCET, 40 in AIEEE, and 40 in IITJEE entrance, 20 failed in MHTCET and AIEEE, 17 in AIEEE and IITJEE entrance, 15 in MHTCET and IITJEE entrance exam and 5 failed in all three examinations. Find how many students (a) did not flunk in any examination (b) failed in AIEEE or IITJEE entrance.

8. From amongst 2000 literate and illiterate individuals of a town, 70 percent read Marathi newspaper, 50 percent read English newspapers, and 32.5 percent read both Marathi and English newspapers. Find the number of individuals who read

8i) at least one of the newspapers

8ii) neither Marathi and English newspaper

8iii) only one of the newspapers

9) In a hostel, 25 students take tea, 20 students take coffee, 15 students take milk, 10 students take both tea and coffee, 8 students take both milk and coffee. None of them take the tea and milk both and everyone takes at least one beverage, find the number of students in the hostel.

10) There are 260 persons with a skin disorder. If 150 had been exposed to chemical A, 74 to chemical B, and 36 to both chemicals A and B, find the number of persons exposed to  (a) Chemical A but not Chemical B (b) Chemical B but not Chemical A (c) Chemical A or Chemical B.

11) If A = \{ 1,2,3\} write down the power set of A.

12) Write the following intervals in Set Builder Form: (a) (-3,0) (b) [6,12] (c) (6,12] (d) [-23,5)

13) Using Venn Diagrams, represent (a) (A \bigcup B)^{'} (b) A^{'} \bigcup B^{'} (c) A^{'} \bigcap B (d) A \bigcap B^{'}

Regards,

Nalin Pithwa.

The importance of lines and slopes

  1. Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
  2. Bodies falling from rest in a planet’s gravitational field do so in a straight line.
  3. Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=36S3685TG7OYF&keywords=thomas+calculus&qid=1561503390&s=books&sprefix=Thomas+%2Caps%2C259&sr=1-1

or Flipkart:

https://www.flipkart.com/thomas-calculus-1/p/itmebug5kzrnttfj?pid=9789332547278&lid=LSTBOK9789332547278CHN4GH&marketplace=FLIPKART&srno=s_1_23&otracker=AS_Query_OrganicAutoSuggest_2_9&otracker1=AS_Query_OrganicAutoSuggest_2_9&fm=SEARCH&iid=fdc8327b-756c-4f6d-aa10-b45acc900e12.9789332547278.SEARCH&ppt=sp&ppn=sp&ssid=uz7zckp71c0000001561503474614&qH=2488f76736a10369

More questions on applications of derivatives: IITJEE mains maths tutorial

  1. Prove that the minimum value of (a+x)(b+x)/(c+x) for x>-c, is (\sqrt{a-c}+\sqrt{b-c})^{2}.
  2. A cylindrical vessel of volume 25\frac{1}{7} cubic meters, open at the top, is to be manufactured from a sheet of metal. Find the dimensions of the vessel so that the amount of metal used is the least possible.
  3. Assuming that the petrol burnt in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of c kmph is 3c/2 kmph.
  4. Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
  5. Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
  6. Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
  7. Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
  8. Break up the number 8 into two parts such that the sum of their cubes is the least possible. (Use calculus techniques, not intelligent guessing).
  9. Find the greatest and least possible values of the following functions on the given interval: (i) y=x+2\sqrt{x} on [0,4]. (ii) y=\sqrt{100-x^{2}} on [-6,8] (iii) y=\frac{a^{2}}{x}+\frac{b^{2}}{1-x} on (0,1) with a>0 and b>0 (iv) y=2\tan{x}-\tan^{2}{x} on [0,\pi/2) (iv) y=\arctan{\frac{1-x}{1+x}} on [0,1].
  10. Prove the following inequalities: (in these subset of questions, you can try to use pure algebraic methods, apart from calculus techniques to derive alternate solutions): (i) 2\sqrt{x} >3-\frac{1}{x} for x>1 (ii) 2x\arctan{x} \geq \log{(1+x^{2})} (iii) \sin{x} < x-\frac{x^{3}}{6}+\frac{x^{5}}{120} for x>0. (iv) \log{(1+x)}>\frac{\arctan{x}}{1+x} for x>0 (iv) e^{x}+e^{-x} > 2+x^{2} for x \neq 0.
  11. Find the interval of monotonicity of the following functions: (i) y=x-e^{x} (ii) y=\log{(x+\sqrt{1+x^{2}})} (iii) y=x\sqrt{ax-x^{2}} (iv) y=\frac{10}{4x^{3}-9x^{2}+6x}
  12. Prove that if 0<x_{1}<x_{2}<\frac{\pi}{2}, then \frac{\tan{x_{2}}}{\tan{x_{1}}} > \frac{x_{2}}{x_{1}}
  13. On the graph of the function y=\frac{3}{\sqrt{2}}x\log{x} where x \in [e^{-1.5}, \infty), find the point M(x,y) such that the segment of the tangent to the graph of the function at the point intercepted between the point M and the y-axis, is the shortest.
  14. Prove that for 0 \leq p \leq 1 and for any positive a and b, the inequality (a+b)^{p} \leq a^{p}+b^{p} is valid.
  15. Given that f^{'}(x)>g^{'}(x) for all real x, and f(0)=g(0), prove that f(x)>g(x) for all x \in (0,\infty), and that f(x)<g(x) for all x \in (-\infty, 0).
  16. If f^{''}(x)<0 for all x \in (a,b), prove that f^{'}(x)=0 at most once in (a,b).
  17. Suppose that a function f has a continuous second derivative, f(0)=0, f^{'}(0)=0, f^{''}(x)<1 for all x. Show that |f(x)|<(1/2)x^{2} for all x.
  18. Show that x=\cos{x} has exactly one root in [0,\frac{\pi}{2}].
  19. Find a polynomial P(x) such that P^{'}(x)-3P(x)=4-5x+3x^{2}. Prove that there is only one solution.
  20. Find a function, if possible whose domain is [-3,3], f(-3)=f(3)=0, f(x) \neq 0 for all x \in (-3,3), f^{'}(-1)=f^{'}(1)=0, f^{'}(x)>0 if |x|>1 and f^{'}(x)<0, if |x|<1.
  21. Suppose that f is a continuous function on its domain [a,b] and f(a)=f(b). Prove that f has at least one critical point in (a,b).
  22. A right circular cone is inscribed in a sphere of radius R. Find the dimensions of the cone, if its volume is to be maximum.
  23. Estimate the change in volume of right circular cylinder of radius R and height h when the radius changes from r_{0} to r_{0}+dr and the height does not change.
  24. For what values of a, m and b does the function: f(x)=3, when x=0; f(x)=-x^{2}+3x+a, when 0<x<1; and f(x)=mx+b, when 1 \leq x \leq 2 satisfy the hypothesis of the Lagrange’s Mean Value Theorem.
  25. Let f be differentiable for all x, and suppose that f(1)=1, and that f^{'}<0 on (-\infty, 1) and that f^{'}>0 on (1,\infty). Show that f(x) \geq 1 for all x.
  26. If b, c and d are constants, for what value of b will the curve y=x^{3}+bx^{2}+cx+d have a point of inflection at x=1?
  27. Let f(x)=1+4x-x^{2} \forall x \in \Re and g(x)= \left\{  \begin{array}{ll} max \{ f(x): x\leq t\leq x+3\} & 0 \leq x \leq 3\\ min (x+3) & 3 \leq x \leq 5 \end{array} \right. Find the critical points of g on [0,5]
  28. Find a point P on the curve x^{2}+4y^{2}-4=0 so that the area of the triangle formed by the tangent at P and the co-ordinate axes is minimum.
  29. Let f(x) = \left \{ \begin{array}{ll} -x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3b+2} & 0 \leq x <1 \\ 2x-3 & 1 \leq x \leq 3 \end{array}\right. Find all possible real values of b such that f(x) has the smallest value at x=1.
  30. The circle x^{2}+y^{2}=1 cuts the x-axis at P and Q. Another circle with centre Q and variable radius intersects the first circle at R above the x-axis and line segment PQ at x. Find the maximum area of the triangle PQR.
  31. A straight line L with negative slope passes through the point (8,2) and cuts the positive co-ordinate axes at points P and Q. Find the absolute minimum value of OP+PQ, as L varies, where O is the origin.
  32. Determine the points of maxima and minima of the function f(x)=(1/8)\log{x}-bx+x^{2}, with x>0, where b \geq 0 is a constant.
  33. Let (h,k) be a fixed point, where h>0, k>0. A straight line passing through this point cuts the positive direction of the co-ordinate axes at points P and Q. Find the minimum area of the triangle OPQ, O being the origin.
  34. Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.
  35. Show that the following functions have at least one zero in the given interval: (i) f(x)=x^{4}+3x+1, with [-2,-1] (ii) f(x)=x^{3}+\frac{4}{x^{2}}+7 with (-\infty,0) (iii) r(\theta)=\theta + \sin^{2}({\theta}/3)-8, with (-\infty, \infty)
  36. Show that all points of the curve y^{2}=4a(x+\sin{(x+a)}) at which the tangent is parallel to axis of x lie on a parabola.
  37. Show that the function f defined by f(x)=|x|^{m}|x-1|^{n}, with x \in \Re has a maximum value \frac{m^{m}n^{n}}{(m+n)^{m+n}} with m,n >0.
  38. Show that the function f defined by f(x)=\sin^{m}(x)\sin(mx)+\cos^{m}(x)\cos(mx) with x \in \Re has a minimum value at x=\pi/4 which m=2 and a maximum at x=\pi/4 when m=4,6.
  39. If f^{''}(x)>0 for all x \in \Re, then show that f(\frac{x_{1}+x_{2}}{2}) \leq (1/2)[f(x_{1})+f(x_{2})] for all x_{1}, x_{2}.
  40. Prove that (e^{x}-1)>(1+x)\log(1+x), if x \in (0,\infty).

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.

 

 

 

Derivatives: IITJEE Mains Maths: Mathematics Hothouse video lecture

Limits Part 2: IITJEE Mains maths: Mathematics Hothouse

Limits part 1: video lecture: IITJEE Mains maths: Mathematics Hothouse

 

Applications of Derivatives: Training for IITJEE Maths: Part V

Question I:

Show that the equation of the tangent to the curve x=a \frac{f(t)}{h(t)} and y=a \frac{g(t)}{h(t)} can be represented in the form:

\left| \begin{array}{ccc} x & y & a \\ f(t) & g(t) & h(t) \\ f^{'}(t) & g^{'}(t) & h^{'}(t) \end{array} \right|=0

Question 2:

Show that the derivative of the function f(x) = x \sin{\frac{\pi}{x}}, when x>0 and f(x)=0, when x=0 vanishes on an infinite set of points of the interval $latex (0,1), Hint: Use Rolle’s theorem.

Question 3:

Prove that \frac{1}{1+x}<\log (1+x) < x for x>0. Use Lagrange’s theorem.

Question 4:

Find the largest term in the sequence a_{n}=\frac{n^{2}}{n^{3}+200}. Hint: Consider the function f(x)=\frac{x^{2}}{x^{3}+200} in the interval [1,\infty).

Question 5:

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Question 6:

Find the polynomial f(x) of degree 6, which satisfies \lim_{x \rightarrow 0} (1+\frac{f(x)}{x^{3}})=e^{2} and has a local maximum at x=1 and local minimum at x=0 and 2.

Question 7:

For the circle x^{2}+y^{2}=r^{2}, find the value of r for which the area enclosed by the tangents drawn from the point (6,8) to the circle and the chord of contact is maximum.

Question 8:

Suppose that f has a continuous derivative for all values of x and f(0)=0, with |f^{'}(x)| <1 for all x. Prove that |f(x)| \leq |x|.

Question 9:

Show that (e^{x}-1)>(1+x)\log {1+x}, if x \in (0,\infty).

Question 10:

Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.

 

Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

Question 1.

If the point on y = x \tan {\alpha} - \frac{ax^{2}}{2u^{2}\cos^{2}{\alpha}}, where \alpha>0, where the tangent is parallel to y=x has an ordinate \frac{u^{2}}{4a}, then what is the value of \alpha?

Question 2:

Prove that the segment of the tangent to the curve y=c/x, which is contained between the coordinate axes is bisected at the point of tangency.

Question 3:

Find all the tangents to the curve y = \cos{(x+y)} for -\pi \leq x \leq \pi that are parallel to the line x+2y=0.

Question 4:

Prove that the curves y=f(x), where f(x)>0, and y=f(x)\sin{x}, where f(x) is a differentiable function have common tangents at common points.

Question 5:

Find the condition that the lines x \cos{\alpha} + y \sin{\alpha} = p may touch the curve (\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1.

Question 6:

Find the equation of a straight line which is tangent to one point and normal to the point on the curve y=8t^{3}-1, and x=4t^{2}+3.

Question 7:

Three normals are drawn from the point (c,0) to the curve y^{2}=x. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

Question 8:

If p_{1} and p_{2} are lengths of the perpendiculars from origin on the tangent and normal to the curve x^{2/3} + y^{2/3}=a^{2/3} respectively, prove that 4p_{1}^{2} + p_{2}^{2}=a^{2}.

Question 9:

Show that the curve x=1-3t^{2}, and y=t-3t^{3} is symmetrical about x-axis and has no real points for x>1. If the tangent at the point t is inclined at an angle \psi to OX, prove that 3t= \tan {\psi} +\sec {\psi}. If the tangent at P(-2,2) meets the curve again at Q, prove that the tangents at P and Q are at right angles.

Question 10:

Find the condition that the curves ax^{2}+by^{2}=1 and a^{'}x^{2} + b^{'}y^{2}=1 intersect orthogonality and hence show that the curves \frac{x^{2}}{(a^{2}+b_{1})} + \frac{y^{2}}{(b^{2}+b_{1})} = 1 and \frac{x^{2}}{a^{2}+b_{2}} + \frac{y^{2}}{(b^{2}+b_{2})} =1 also intersect orthogonally.

More later,

Nalin Pithwa.