Category Archives: IITJEE Foundation mathematics

Set theory, relations, functions: preliminaries: Part V

Types of functions: (please plot as many functions as possible from the list below; as suggested in an earlier blog, please use a TI graphing calculator or GeoGebra freeware graphing software): 

  1. Constant function: A function f:\Re \longrightarrow \Re given by f(x)=k, where k \in \Re is a constant. It is a horizontal line on the XY-plane.
  2. Identity function: A function f: \Re \longrightarrow \Re given by f(x)=x. It maps a real value x back to itself. It is a straight line passing through origin at an angle 45 degrees to the positive X axis.
  3. One-one or injective function: If different inputs give rise to different outputs, the function is said to be injective or one-one. That is, if f: A \longrightarrow B, where set A is domain and set B is co-domain, if further, x_{1}, x_{2} \in A such that x_{1} \neq x_{2}, then it follows that f(x_{1}) \neq f(x_{2}). Sometimes, to prove that a function is injective, we can prove the conrapositive statement of the definition also; that is, y_{1}=y_{2} where y_{1}, y_{2} \in codomain \hspace{0.1in} range, then it follows that x_{1}=x_{2}. It might be easier to prove the contrapositive. It would be illuminating to construct your own pictorial examples of such a function. 
  4. Onto or surjective: If a function is given by f: X \longrightarrow Y such that f(X)=Y, that is, the images of all the elements of the domain is full of set Y. In other words, in such a case, the range is equal to co-domain. it would be illuminating to construct your own pictorial examples of  such a function.
  5. Bijective function or one-one onto correspondence: A function which is both one-one and onto is called a bijective function. (It is both injective and surjective). Only a bijective function will have a well-defined inverse function. Think why! This is the reason why inverse circular functions (that is, inverse trigonometric functions have their domains restricted to so-called principal values). 
  6. Polynomial function: A function of the form f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots + a_{n}x^{n}, where n is zero or positive integer only and a_{i} \in \Re is called a polynomial with real coefficients. Example. f(x)=ax^{2}=bx+c, where a \neq 0, a, b, c \in \Re is called a quadratic function in x. (this is a general parabola).
  7. Rational function: The function of the type \frac{f(x)}{g(x)}, where g(x) \neq 0, where f(x) and g(x) are polynomial functions of x, defined in a domain, is called a rational function. Such a function can have asymptotes, a term we define later. Example, y=f(x)=\frac{1}{x}, which is a hyperbola with asymptotes X and Y axes.
  8. Absolute value function: Let f: \Re \longrightarrow \Re be given by f(x)=|x|=x when x \geq 0 and f(x)=-x, when x<0 for any x \in \Re. Note that |x|=\sqrt{x^{2}} since the radical sign indicates positive root of a quantity by convention.
  9. Signum function: Let f: \Re \longrightarrow \Re where f(x)=1, when x>0 and f(x)=0 when x=0 and f(x)=-1 when x<0. Such a function is called the signum function. (If you can, discuss the continuity and differentiability of the signum function). Clearly, the domain of this function  is full \Re whereas the range is \{ -1,0,1\}.
  10. In part III of the blog series, we have already defined the floor function and the ceiling function. Further properties of these functions are summarized (and some with proofs in the following wikipedia links): (once again, if you can, discuss the continuity and differentiablity of the floor and ceiling functions): https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
  11. Exponential function: A function f: \Re \longrightarrow \Re^{+} given by f(x)=a^{x} where a>0 is called an exponential function. An exponential function is bijective and its inverse is the natural logarithmic function. (the logarithmic function is difficult to define, though; we will consider the details later). PS: Quiz: Which function has a faster growth rate — exponential or a power function ? Consider various parameters.
  12. Logarithmic function: Let a be a positive real number with a \neq 1. If a^{y}=x, where x \in \Re, then y is called the logarithm of x with base a and we write it as y=\ln{x}. (By the way, the logarithmic function is used in the very much loved mp3 music :-))

Regards,

Nalin Pithwa

Set Theory, Relations and Functions: Preliminaries: IV:

Problem Set based on previous three parts:

I) Solve the inequalities in the following exercises expressing the solution sets as intervals or unions of intervals. Also, graph each solution set on the real line:

a) |x| <2 (b) |x| \leq 2 (c) |t-1| \leq 3 (d) |t+2|<1 (e) |3y-7|<4(f) |2y+5|<1 (g) |\frac{z}{5}-1| \leq 1 (h) | \frac{3}{2}z-1| \leq 2 (i) |3-\frac{1}{x}|<\frac{1}{2} (j) |\frac{2}{x}-4|<3 (k) |2x| \geq 4 (l) |x+3| \geq \frac{1}{2} (m) |1-x| >1 (n) |2-3x| > 5 (o) |\frac{x+1}{2}| \geq 1 (p) |\frac{3x}{5}-1|>\frac{2}{5}

II) Quadratic Inequalities:

Solve the inequalities in the following exercises. Express the solution sets as intervals or unions of intervals and graph them. Use the result \sqrt{a^{2}}=|a| as appropriate.

(a) x^{2}<2 (b) 4 \leq x^{2} (c) 4<x^{2}<9 (d) \frac{1}{9} < x^{2} < \frac{1}{4} (e) (x-1)^{2}<4 (f) (x+3)^{2}<2 (g) x^{2}-x<0 (h) x^{2}-x-2 \geq 0

III) Theory and Examples:

i) Do not fall into the trap |-a|=a. For what real numbers a is the equation true? For what real numbers is it false?

ii) Solve the equation: |x-1|=1-x.

iii) A proof of the triangle inequality: 

Give the reason justifying each of the marked steps in the following proof of the triangle inequality:

|a+b|^{2}=(a+b)^{2}…..why ?

=a^{2}+2ab++b^{2}

\leq a^{2}+2|a||b|+b^{2}….why ?

\leq |a|^{2}+2|a||b|+|b|^{2}….why?

=(|a|+|b|)^{2}….why ?

iv) Prove that |ab|=|a||b| for any numbers a and b.

v) If |x| \leq 3 and x>-\frac{1}{2}, what can you say about x?

vi) Graph the inequality: |x|+|y| \leq 1

Questions related to functions:

I) Find the domain and range of each function:

a) f(x)=1-\sqrt{x} (b) F(t)=\frac{1}{1+\sqrt{t}} (c) g(t)=\frac{1}{\sqrt{4-t^{2}}}

II) Finding formulas for functions:

a) Express the area and perimeter of an equilateral triangle as a function of the triangle’ s side with length s.

b) Express the side length of a square as a function of the cube’s diagonal length d. Then, express the surface area  and volume of the cube as a function of the diagonal length.

c) A point P in the first quadrant lies on the graph of the function f(x)=\sqrt{x}. Express the coordinates of P as functions of the slope of the line joining P to the origin.

III) Functions and graphs:

Graph the functions in the questions below. What symmetries, if any, do the graphs have?

a) y=-x^{3} (b) y=-\frac{1}{x^{2}} (c) y=-\frac{1}{x} (d) y=\frac{1}{|x|} (e) y = \sqrt{|x|} (f) y=\sqrt{-x} (g) y=\frac{x^{3}}{8} (h) y=-4\sqrt{x} (i) y=-x^{\frac{3}{2}} (j) y=(-x)^{\frac{3}{2}} (k) y=(-x)^{\frac{2}{3}} (l) y=-x^{\frac{2}{3}}

IV) Graph the following equations ad explain why they are not graphs of functions of x. (a) |y|=x (b) y^{2}=x^{2}

V) Graph the following equations and explain why they are not graphs of functions of x: (a) |x|+|y|=1 (b) |x+y|=1

VI) Even and odd functions:

In the following questions, say whether the function is even, odd or neither.

a) f(x)=3 (b) f(x=x^{-5} (c) f(x)=x^{2}+1 (d) f(x)=x^{2}+x (e) g(x)=x^{4}+3x^{2}-1 (f) g(x)=\frac{1}{x^{2}-1} (g) g(x)=\frac{x}{x^{2}-1} (h) h(t)=\frac{1}{t-1} (i) h(t)=|t^{3}| (j) h(t)=2t+1 (k) h(t)=2|t|+1

Sums, Differences, Products and Quotients:

In the two questions below, find the domains and ranges of f, g, f+g, and f-g.

i) f(x)=x, g(x)=\sqrt{x-1} (ii) f(x)=\sqrt{x+1}, g(x)=\sqrt{x-1}

In the two questions below, find the domains and ranges of f, g, \frac{f}{g} and \frac{g}{f}

i) f(x)=2, g(x)=x^{2}+1

ii) f(x)=1, g(x)=1+\sqrt{x}

Composites of functions:

  1. If f(x)=x+5, and g(x)=x^{2}-5, find the following: (a) f(g(0)) (b) g(f(0)) (c) f(g(x)) (d) g(f(x)) (e) f(f(-5)) (f) g(g(2)) (g) f(f(x)) (h) g(g(x))
  2. If f(x)=x-1 and g(x)=\frac{1}{x+1}, find the following: (a) f(g(\frac{1}{2})) (b) g(f(\frac{1}{2})) (c) f(g(x)) (d) g(f(x)) (e) f(f(2)) (f) g(g(2)) (g) f(f(x)) (h) g(g(x))
  3. If u(x)=4x-5, v(x)=x^{2}, and f(x)=\frac{1}{x}, find formulas or formulae for the following: (a) u(v(f(x))) (b) u(f(v(x))) (c) v(u(f(x))) (d) v(f(u(x))) (e) f(u(v(x))) (f) f(v(u(x)))
  4. If f(x)=\sqrt{x}, g(x)=\frac{x}{4}, and h(x)=4x-8, find formulas or formulae for the following: (a) h(g(f(x))) (b) h(f(g(x))) (c) g(h(f(x))) (d) g(f(h(x))) (e) f(g(h(x))) (f) f(h(g(x)))

Let f(x)=x-5, g(x)=\sqrt{x}, h(x)=x^{3}, and f(x)=2x. Express each of the functions in the questions below as a composite involving one or more of f, g, h and j:

a) y=\sqrt{x}-3 (b) y=2\sqrt{x} (c) y=x^{\frac{1}{4}} (d) y=4x (e) y=\sqrt{(x-3)^{3}} (f) y=(2x-6)^{3} (g) y=2x-3 (h) y=x^{\frac{3}{2}} (i) y=x^{9} (k) y=x-6 (l) y=2\sqrt{x-3} (m) \sqrt{x^{3}-3}

Questions:

a) g(x)=x-7, f(x)=\sqrt{x}, find (f \circ g)(x)

b) g(x)=x+2, f(x)=3x, find (f \circ g)(x)

c) f(x)=\sqrt{x-5}, (f \circ g)(x)=\sqrt{x^{2}-5}, find g(x).

d) f(x)=\frac{x}{x-1}, g(x)=\frac{x}{x-1}, find (f \circ g)(x)

e) f(x)=1+\frac{1}{x}, (f \circ g)(x)=x, find g(x).

f) g(x)=\frac{1}{x}, (f \circ g)(x)=x, find f(x).

Reference: Calculus and Analytic Geometry, G B Thomas. 

NB: I have used an old edition (printed version) to prepare the above. The latest edition may be found at Amazon India link:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=1XDE2XDSY5LCP&keywords=gb+thomas+calculus&qid=1570492794&s=books&sprefix=G+B+Th%2Caps%2C255&sr=1-1

Regards,

Nalin Pithwa

 

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: 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.

IITJEE Foundation Maths: Variation

DEFINITION:

One quantity A is said to vary directly as another B, when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio.

NOTE: The word directly is often omitted, and A is said to vary as B.

For instance: if a train a moving at a uniform rate travels 40 miles in 60 minutes, it will travel 20 miles in 30 minutes, 80 miles in 120 minutes, and so on; the distance in each case being increased or diminished in the same ratio as the time. This is expressed by saying that when the velocity is uniform the distance is proportional to the time, or the distance varies as the time.

NOTATION: The symbol \alpha is used to denote variation; so that, A \alpha B is read as “A varies as B.”

Theorem I: If A varies as B, then A is equal to B multiplied by some constant quantity.

Note: If any pair of corresponding values of A and B are known, the constant m can be determined. For instance, if A=B, when B=12, we have 3=m \times 12; and m=\frac{1}{4}, and A=\frac{1}{4}B

DEFINITION: One quantity A is said to vary inversely as another B, when A varies directly as the reciprocal of B.

The following is an illustration of inverse variation: If 6 men do a certain work in 8 hours, 12 men would do the same work in 4 hours; 2 men in 24 hours; and so on. Thus, it appears that when the number of men is increased, the same is proportionately decreased; and vice-versa.

Example 1: The cube root of x varies inversely as the square of y; if x=8, when y=3; find x when y=\frac{3}{2}.

Solution 1: By supposition, \sqrt[3]{x}=\frac{m}{y^{2}}, where m is constant. Putting x=8, y=3, we have 2=\frac{m}{9}, so m=18, and \sqrt[3]{x}=\frac{18}{y^{2}}; hence, by putting y=\frac{3}{2}, we obtain x=512.

Example 2: The square of the time of a planet’s revolution varies as the cube of its distance from the Sun; find the time of Venus’s revolution, assuming the distances of the Earth and Venus from the Sun to be 91\frac{1}{4} and 66 millions of miles respectively.

Let P be the periodic time measured in days, D the distance in millions of miles; we have

P^{2} \alpha D^{3}, or P^{2}=k \times D^{3}, where k is some constant.

For the Earth, 365 \times 365 = k \times 91 \frac{1}{4} \times 91 \frac{1}{4} \times 91 \times \frac{1}{4}

hence, k=\frac{4 \times 4 \times 4}{3.5}

so that P^{2}=\frac{4 \times 4 \times 4}{365}D^{3}

For Venus, P^{2}=\frac{4 \times 4 \times 4}{365} \times 66 \times 66 \times 66

hence, P= A \times 66 \times \sqrt{\frac{264}{365}} = 264 \times \sqrt{0.7233}, approximately.

P^{2}=264 \times 0.85=224.4.

Hence, the time of revolution is nearly 224\frac{1}{2}.

DEFINITION: One quantity is said to vary jointly as a number of others, when it varies directly as their product.

Thus, A varies jointly as B and C, when A=m \times BC. For instance, the interest on a sum of money varies jointly as the principal, the time, and the rate per cent.

DEFINITION: 

A is said to vary directly as B and inversely as C, when A varies as \frac{B}{C}.

Theorem:

If A varies as B, when C is constant, and A varies as C when B is constant, then A will vary as BC when both B and C vary.

Proof:

Homework

The following are some illustrations of the theorems stated above:

The amount of work done by a given number of men varies directly as the number of days they work, and the amount of work done in a given time varies directly as the number of men; therefore, when the number of days and the number of men are both variable, the amount of work will vary as the product of the number of men and the number of days.

Again, in plane geometry, the area of a triangle varies directly as its base when the height is constant, and directly as the height when the base is constant; and when both the height and base are variable, the area varies as the product of the numbers representing the height and the base.

Example:

The volume of a right circular cone varies as the square of the radius of the base when the height is constant, and as the height when the base is constant. If the radius of base is 7cm, and the height is 15 cm, the volume is 770 cc, find the height of a cone whose volume is 132 cubic cm, and which stands on a base whose radius is 3cm.

Solution:9 

Let h and r denote respectively the height and radius of the base measured in cm.; also let V be the volume in cubic cm.

Then. V=m \times r^{2} \times h, where m is constant.

By assumption, 770=m \times 7^{2} \times 15

hence, m = \frac{22}{21} so that V=\frac{22}{21}r^{2}h.

By substituting V=132, r=3, we get the following:

132=\frac{22}{21} \times 9 \times h

so that h=14; and, therefore the height is 14 cm.

Note:

A quantity A can vary jointly as (a product of) more than two variables also as is most often the case in real engineering. Further, the variations may be either direct or inverse. The principle is interesting because of its frequent occurence in physical sciences or engineering. For example, Boyle’s law in chemistry: It is found by experiment that the pressure (P) of a gas varies as the “absolute temperature” (T) (in Kelvin) when its volume (V) is constant and that the pressure varies inversely as the volume when the temperature is constant; that is,

P \alpha T, when V is constant and P \alpha \frac{1}{V} when T is constant.

From these results we should expect that, when both t and v are variable, we should have the formula:

P \alpha \frac{T}{V}, or PV=kT, where k is a constant (based on laws of chemistry). And, by actual experiment this is found to be true.

Example:

The duration of a railway journey varies ditectly as the distance and inversely as the velocity; the velocity varies directly as the square root of the quantity of coal used per kilometer (don’t worry the days of steam engine/coal engine and resulting environmental degradation are over; but this is only a simple engineering application), and inversely as the number of carriages in the train. In a journey of 50 kilometers, in half an hour with 18 carriages 100 kg of coal is required; how much coal will be consumed in a journey of 42 kilometers, in 28 minutes with 16 carriages?

Solution:

Let t be the time expressed in hours; let d be the distance in kilometers; let v be the velocity in kmph; let q be the mass of coal (in kg) used per kilometers; and let c be the number of carriages.

We have t \alpha \frac{d}{v} and v \alpha \frac{\sqrt{q}}{c}, and hence, t \alpha \frac{cd}{\sqrt{q}}, or t=\frac{kcd}{\sqrt{q}}, where k is a constant.

Substituting the values given, we have (since q=2),

\frac{1}{2} = \frac{k \times 18 \times 50}{\sqrt{2}}

that is, k=\frac{1}{\sqrt{2} \times 18 \times 50}.

Hence, t=\frac{cd}{\sqrt{2} \times 18 \times 50\sqrt{q}}

Substituting now the values of t, c, d given in the second part of the question, we have

\frac{28}{60}=\frac{16 \times 42}{\sqrt{2} \times 18 \times 50 \times \sqrt{q} }

that is, \sqrt{q} = 4 \sqrt{2}, hence q=32.

Hence, the quantity of coal is 42 \times 32 = 1344 kg.

Tutorial problems on Variation:

  1. If x varies as y, and x=8, when y=15, find x when y=10
  2. If P varies as Q, and P=7 when Q=3, find P when Q=2\frac{1}{3}.
  3.  If the square of x varies as the cube of y, and x=3, when y=4, find the value of y when x=\frac{1}{\sqrt{3}}.
  4. A varies as B and C jointly; if A=2 when B=\frac{3}{5} and C=\frac{10}{27}, find C when A=54 and B=3.
  5. If A varies as C, and B varies as C, then A \pm B and \sqrt{AB} will each vary as C.
  6. If A varies as BC, then B varies inversely as \frac{C}{A}.
  7. P varies directly as Q and inversely as R; also P=\frac{2}{3} when Q=\frac{3}{7} and R=\frac{9}{14}; find Q when P=\sqrt{48} and R=\sqrt{75}.
  8. If x varies as y, prove that x^{2}+y^{2} varies as x^{2}-y^{2}.
  9. If y varies as the sum of two quantities, of which one varies directly as x and the other inversely as x; and if y=6 then x=4, and y=3\frac{1}{3} when x=3, find the equation between x and y.
  10. If y is equal to the sum of two quantities one of which varies as x directly, and the other as x^{2} inversely; and, if y=19 when x=2, or 3; find y in terms of x.
  11. If A varies directly as the square root of B and inversely as the cube of C, and if A=3, when B=256 and C=2, find B when A=24 and C = \frac{1}{2}
  12. Given that x+y varies as x+\frac{1}{x}, and that x-y varies as z- \frac{1}{z}, find the relation between x and z, provided that z=2 when x=3 and y=1.
  13. If A varies as B and C jointly, write B varies as D^{2}, and C varies inversely as A, show that A varies as D.
  14. If y varies as the sum of three quantities of which the first is a constant, the second varies as x, and the third as x^{2}; and, if y=0 when x=1, y=1, when x=2, and y=4 when x=3; find y when x=7.
  15. When a body falls down from rest the distance from the starting point varies as the square of the time it has been falling; if a body falls through 122.6 meters in 5 seconds, how far does it fall in 10 seconds? Also, how far does it fall in the tenth second?
  16. Given that the volume of a sphere varies as the cube of its radius, and that when the radius is 3.5 cm, the volume is 176.7 cubic cm; find the volume when the radius is 1.75 cm.
  17. The weight of a circular disc varies as the square of the radius when the thickness remains the same; it also varies as the thickness when the radius remains the same. Two discs have their thicknesses in the ratio of 9:8; find the ratio of their radii if the weight of the first is twice that of the second.
  18. At a certain regatta, the numbers of races on each day varied jointly as the number of days from the beginning and end of the regatta up to and including the day in question. On three successive days there were respectively 6, 5 and 3 races. Which days were these, and how long did the regatta last?
  19. The price of a diamond varies as the square of its weight (mass). Three rings of equal weight, each composed of a diamond set in gold, have values INR a, INR b, INR c, the diamonds in them weighing 3, 4, 5 carats respectively. Show that the value of a diamond of one carat is INR (\frac{a+c}{2}-b), the cost of workmanship being the same for each ring.
  20. Two persons are awarded pensions in proportion to the square root of the number of root of the number of years they have served. One has served 9 years longer than the other and receives a pension greater by INR 500. If the length of service of the first had exceeded that of the second by 4\frac{1}{4} years their pensions would have been in the proportion of 9:8. How long had they served and what were their respective pensions?
  21. The attraction of a planet on the satellites varies directly as the mass (M) of the planet, and inversely as the square of the distance (D); also the square of a satellite’s time of revolution varies directly as the distance and inversely as the force of attraction. If m_{1}, d_{1}, t_{1} and m_{2}, d_{2}, t_{2} are simultaneous values of M, D, T respectively, prove that \frac{m_{1}t_{1}^{2}}{m_{2}t_{2}^{2}} = \frac{d_{1}^{3}}{d_{2}^{3}}. Hence, find the time of revolution of that moon of Jupiter whose distance is to the distance of our Moon as 35:31, having given that the mass of Jupiter is 343 times that of the Earth, and that the Moon’s period is 27.32 days.
  22. The consumption of coal by a locomotive varies as the square of the velocity; when the speed is 32 kmph the consumption of coal per hour is 2 tonnes: if the price of coal is INR 10 per tonne, and the other expenses of the engine be INR 11.25 an hour, find the least cost of a journey of 100 km.

Cheers,

Nalin Pithwa

 

 

 

 

 

 

 

 

 

 

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…

https://www.cs.virginia.edu/~robins/YouAndYourResearch.html

 

Elementary algebra: fractions: IITJEE foundation maths

Expertise in dealing with algebraic fractions is necessary especially for integral calculus, which is of course, a hardcore area of IITJEE mains or advanced maths.

Below is a problem set dealing with fractions; the motivation is to develop super-speed and super-fine accuracy:

A) Find the value of the following: (the answer should be in as simple terms as possible, which means, complete factorization will be required):

1) \frac{1}{6a^{2}+54} + \frac{1}{3a-9} - \frac{a}{3a^{2}-27}

2) \frac{1}{6a-18} - \frac{1}{6a+18} -\frac{1}{a^{2}+9} + \frac{18}{a^{4}+81}

3) \frac{1}{8-8x} - \frac{1}{8+8x} + \frac{x}{4+4x^{2}} - \frac{x}{2+2x^{4}}

4) \frac{x+1}{2x^{3}-4x^{2}} + \frac{x-1}{2x^{3}+4x^{2}} - \frac{1}{x^{2}-4}1

5) \frac{1}{3x^{2}-4xy+y^{2}} + \frac{1}{x^{2}-4xy+3y^{2}} -\frac{3}{3x^{2}-10xy+3y^{2}}

6) \frac{1}{x-1} + \frac{2}{x+1} - \frac{3x-2}{x^{2}-1} - \frac{1}{(x+1)^{2}}

7) \frac{108-52x}{x(3-x)^{2}} - \frac{4}{3-x} - \frac{12}{x} + (\frac{1+x}{3-x})^{2}

8) \frac{(a+b)^{2}}{(x-a)(x+a+b)} - \frac{a+2b+x}{2(x-a)} + \frac{(a+b)x}{x^{2}+bx-a^{2}-ab} + \frac{1}{2}

9) \frac{3(x^{2}+x-2)}{x^{2}-x-2} -\frac{3(x^{2}-x-2)}{x^{2}+x-2} - \frac{8x}{x^{2}-4}

More later,

Nalin Pithwa

References for IITJEE Foundation Mathematics and Pre-RMO (Homi Bhabha Foundation/TIFR)

  1. Algebra for Beginners (with Numerous Examples): Isaac Todhunter (classic text): Amazon India link: https://www.amazon.in/Algebra-Beginners-Isaac-Todhunter/dp/1357345259/ref=sr_1_2?s=books&ie=UTF8&qid=1547448200&sr=1-2&keywords=algebra+for+beginners+todhunter 
  2. Algebra for Beginners (including easy graphs): Metric Edition: Hall and Knight Amazon India link: https://www.amazon.in/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=algebra+for+beginners+hall+and+knight
  3. Elementary Algebra for School: Metric Edition: https://www.amazon.in/Elementary-Algebra-School-H-Hall/dp/8185386854/ref=sr_1_5?s=books&ie=UTF8&qid=1547448497&sr=1-5&keywords=elementary+algebra+for+schools
  4. Higher Algebra: Hall and Knight: Amazon India link: https://www.amazon.in/Higher-Algebra-Knight-ORIGINAL-MASPTERPIECE/dp/9385966677/ref=sr_1_6?s=books&ie=UTF8&qid=1547448392&sr=1-6&keywords=algebra+for+beginners+hall+and+knight
  5. Plane Trigonometry: Part I: S L Loney: https://www.amazon.in/Plane-Trigonometry-Part-1-S-L-Loney/dp/938592348X/ref=sr_1_16?s=books&ie=UTF8&qid=1547448802&sr=1-16&keywords=plane+trigonometry+part+1+by+s.l.+loney

The above references are a must. Best time to start is from standard VII or standard VIII.

-Nalin Pithwa.

Binomial Theorem: part 2: algebra for math olympiads or IITJEE maths

Binomial Theorem: Part I: A Primer for mathematical olympiads and IITJEE/RMO