Regards,
Nalin Pithwa
[contact-form] ]]>Regards,
Nalin Pithwa
[contact-form] ]]>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) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)
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 as appropriate.
(a) (b) (c) (d) (e) (f) (g) (h)
III) Theory and Examples:
i) Do not fall into the trap . For what real numbers a is the equation true? For what real numbers is it false?
ii) Solve the equation: .
iii) A proof of the triangle inequality:
Give the reason justifying each of the marked steps in the following proof of the triangle inequality:
…..why ?
….why ?
….why?
….why ?
iv) Prove that for any numbers a and b.
v) If and , what can you say about x?
vi) Graph the inequality:
Questions related to functions:
I) Find the domain and range of each function:
a) (b) (c)
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 . 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) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
IV) Graph the following equations ad explain why they are not graphs of functions of x. (a) (b)
V) Graph the following equations and explain why they are not graphs of functions of x: (a) (b)
VI) Even and odd functions:
In the following questions, say whether the function is even, odd or neither.
a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)
Sums, Differences, Products and Quotients:
In the two questions below, find the domains and ranges of , , , and .
i) , (ii) ,
In the two questions below, find the domains and ranges of , , and
i) ,
ii) ,
Composites of functions:
Let , , , and . Express each of the functions in the questions below as a composite involving one or more of f, g, h and j:
a) (b) (c) (d) (e) (f) (g) (h) (i) (k) (l) (m)
Questions:
a) , , find
b) , , find
c) , , find .
d) , , find
e) , , find .
f) , , find .
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:
Regards,
Nalin Pithwa
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For what values of x, is (a) (b) ?
Problem 2:
Which real numbers x satisfy the equation ?
Problem 3:
Does for all real x? Give reasons for your answer.
Problem 4:
Graph: when ; and , when .
Why is f(x) called the integer part of x? Discuss the continuity and differentiability of f(x).
Cheers,
Nalin Pithwa
[contact-form]]]>
Proof 1:
Let a, b, c be roots of a polynomial P(X) Then by fundamental theorem of algebra
.
Now, once again basic algebra says that as a, b, c are roots/solutions of the above:
=0$
Adding all the above:
So, we get
Also, the above formula can be written as
Proof 2:
Consider the following determinant D:
On adding all three columns to the first column: we know that the value of the determinant is unchanged: So we get the following:
. Note that columns 2 and 3 of the three by three determinant do not change.
On expanding the original determinant D, we get
Whereas we get from the other transformed but equal D:
So, that we again get
Proof 3:
Now, let us consider as a quadratic in a with b and c as parameters.
That is,
Then, the discriminant is given by
, which in turn equals:
Hence, the factorization of the above quadratic in a is given as:
So, the other non-trivial factorization of the above famous algebraic identity is:
where are cube roots of unity.
Proof 4:
Factorize the expression .
Solution 4:
We can carry out the above polynomial division by considering the dividend to be a polynomial in a single variable, say a, (and assuming b and c are just parameters; so visualize them as arbitrary but fixed constants); further arrange the dividend in descending powers of a; so also arrange the divisor in descending powers of a (well, of course, it is just linear in a; and assume b and c are parameters also in dividend).
Proof 5:
Prove that the eliminant of
is
Proof 5:
By Cramer’s rule, the eliminant is given by determinant .
On expansion using the first row:
upon multiplying both the sides of the above equation by (-1). Of course, we have only been able to generate the basic algebraic expression but we have done so by encountering a system of three linear equations in x, y, z. (we could append any of the above factorization methods to this further!! :-)))
Cheers,
Nalin Pithwa
[contact-form] ]]>But, what is most characteristic of his work is its clarity and openness. He never tries to hide the difficulties from the reader. This is in stark contrast to Newton, who was prone to hide his methods in obscure anagrams, and even from his successor, Gauss, who very often erased his steps to present a monolithic proof that was seldom illuminating. In Euler’s writings there are no comments on how profound his results are, and in his papers one can follow his ideas step by step with the greatest of ease. Nor was he chary of giving credit to others; his willingness to share his summation formula with Maclaurin, his proper citations to Fuguano when he started his work on algebraic integrals, his open admiration for Lagrange when the latter improved on his work in calculus of variations are all instances of his serene outlook. One can only contrast this with Gauss’s reaction to Bolyai’s discovery of non-Euclidean postulates. Euler was secure in his knowledge of what he had achieved but never insisted that he should be the only one on top of the mountain.
Perhaps, the most astounding aspect of his scientific opus is its universality. He worked on everything that had any bearing on mathematics. For instance, his early training under Johann Bernoulli did not include number theory; nevertheless, within a couple of years after reaching St. Petersburg he was deeply immersed in it, recreating the entire corpus of Fermat’s work in that area and then moving well beyond him. His founding of graph theory as a separate discipline, his excursions in what we call combinatorial topology, his intuition that suggested to him the idea of exploring multizeta values are all examples of a mind that did not have any artificial boundaries. He had no preferences about which branch of mathematics was dear to him. To him, they were all filled with splendour, or Herrlichkeit, to use his own favourite word.
Hilbert and Poincare were perhaps last of the universalists of modern era. Already von Neumann had remarked that it would be difficult even to have a general understanding of more than a third of the mathematicians of his time. With the explosive growth of mathematics in the twentieth century we may never see again the great universalists. But who is to say what is and is not possible for the human mind?
It is impossible to read Euler and not fall under his spell. He is to mathematics what Shakespeare is to literature and Mozart to music: universal and sui generis.
Reference:
Euler Through Time: A New Look at Old Themes by V S Varadarajan:
Hindustan Book Agency;
http://www.hindbook.com/index.php/euler-through-time-a-new-look-at-old-themes;
Amazon India link:
[contact-form]
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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.
[contact-form] ]]>Functions as a special kind of relation:
Let us first consider an example where set A is a set of men, and B is a set of positive real numbers. Let us say f is a relation from A to B given by :
Hence, f “relates” every man in set A to his weight in set B. That is,
i) Every man has some weight associated with him in set B. (ii) That weight is unique. That is, a person cannot have more than one weight (at a given time, of course) !! This, of course, does not mean that two different persons, say P and Q may not have the same weight 100 kg ( the same element of set B). The only thing it means is that any one person, say P will have one and only one weight (100kg) at the time instant of measurement and not more than one weights (which would be crazy) at a time instant it is measured !!
Definition I (a function defined as a relation):
A function f from a set A (called domain) to a set B (called codomain) is a relation that associates or “pairs up” every element of domain A with a unique element of codomain B. (Note that whereas a relation from a set A to a set B is just a subset of the cartesian product ).
Some remarks: The above definition is also motivated by an example of a function as a relation. On the other hand, another definition of a function can be motivated as follows:
We know that the boiling point of water depends on the height of water above sea level. We also know that the simple interest on a deposit in a bank depends on the duration of deposit held in the bank. In these and several such examples, one quantity, say y, depends on another quantity “x”.
Symbol: ; if , then we also denote: ; we also write , read as “y is f of x”.
Here, y is called image of x under f and x is called the preimage of y under f.
Definition: Range: The set of all images in B is called the range of f. That is,
Note: (i) Every function is a relation but every relation need not be a function. (Homework quiz: find illustrative examples for the same) (ii) If the domain and codomain are not specified, they are assumed to be the set of real numbers.
In calculus, we often want to refer to a generic function without having any particular formula in mind. Leonhard Euler invented a symbolic way to say “y is a function of x” by writing
(“y equals f of x”)
In this equation, the symbol f represents the function. The letter x, called the independent variable, represents an input value from the domain of f, and y, the dependent variable, represents the corresponding output value f(x) in the range of f. Here is the formal definition of function: (definition 2):
A function from a set D to a set is a rule that assigns a unique element f(x) in to each element x in D.
In this definition, D=D(f) (read “D of f”) is the domain of the function f and is the range (or codomain containing the range of f).
Think of a function f as a kind of machine that produces an output value f(x) in its range whenever we feed it an input value x from its domain. In our scope, we will usually define functions in one of two ways:
a) by giving a formula such as that uses a dependent variable y to denote the value of the function, or
b) by giving a formula such as that defines a function symbol f to name the function.
NOTE: there can be well-defined functions which do not have any formula at all; for example, let when and , when .
Strictly speaking, we should call the function f and not f(x) as the latter denotes the value of the function at the point x. However, as is common usage, we will often refer to the function as f(x) in order to name the variable on which f depends.
It is sometimes convenient to use a single letter to denote both a function and the dependent variable. For instance, we might say that the area A of a circle of radius r is given by the function : .
Evaluation:
As we said earlier, most of the functions in our scope will be real-valued function of a real variable, functions whose domains and ranges are sets of real numbers. We evaluate such functions by susbtituting particular values from the domain into the function’s defining rule to calculate the corresponding values in the range.
Example 1:
The volume V of a ball (solid sphere) r is given by the function: .
The volume of a ball of radius 3 meters is : .
Example 2:
Suppose that the function F is defined for all real numbers t by the formula: .
Evaluate F at the output values 0, 2, , and F(2).
Solution 2:
In each case, we substitute the given input value for t into the formula for F:
The Domain Convention
When we define a function with a formula and the domain is not stated explicitly, the domain is assumed to be the largest set of x-values for which the formula gives real x-values. This is the function’s so-called natural domain. If we want the domain to be restricted in some way, we must say so.
The domain of the function is understood to be the entire set of real numbers. The formula gives a real value y-value for every real number x. If we want to restrict the domain to values of x greater than or equal to 2, we must write ” ” for .
Changing the domain to which we apply a formula usually changes the range as well. The range of is . The range of where is the set of all numbers obtained by squaring numbers greater than or equal to 2. In symbols, the range is or or
Example 3:
Function : ; domain ; Range (y) is
Function: ; domain ; Range (y) is
Function: ; domain and range (y) is
Function , domain , and range (y) is
Graphs of functions:
The graph of a function f is the graph of the equation . It consists of the points in the Cartesian plane whose co-ordinates are input-output pairs for f.
Not every curve you draw is the graph of a function. A function f can have only one value f(x) for each x in its domain so no vertical line can intersect the graph of a function more than once. Thus, a circle cannot be the graph of a function since some vertical line intersect the circle twice. If a is in the domain of a function f, then the vertical line will intersect the graph of f in the single point .
Example 4: Graph the function over the interval . (homework).Thinking further: so plotting the above graph requires a table of x and y values; but how do we connect the points ? Should we connect two points by a straight line, smooth line, zig-zag line ??? How do we know for sure what the graph looks like between the points we plot? The answer lies in calculus, as we will see in later chapter. There will be a marvelous mathematical tool called the derivative to find a curve’s shape between plotted points. Meanwhile, we will have to settle for plotting points and connecting them as best as we can.
PS: (1) you can use GeoGebra, a beautiful freeware for plotting various graphs, and more stuff https://www.geogebra.org/ (2) If you wish, you can use a TI-graphing calculator. This is a nice investment for many other things like number theory also. See for example,
Meanwhile, you need to be extremely familiar with graphs of following functions; plot and check on your own:
, , , , , , , where ,
Sums, Differences, Products and Quotients
Like numbers, functions can be added, subtracted, multiplied and divided (except where the the denominator is zero) to produce new functions. If f and g are functions, then for every x that belongs to the domains of BOTH f and g, we define functions: , , by the formulas:
,
At any point at which , we can also define the function by the formula:
, where
Functions can also be multiplied by constants. If c is a real number, then the function cf is defined for all x in the domain of f by
Example 5:
Function , formula , domain
Function , formula , domain
Function , formula , domain
Function , formula , domain
Function , formula , domain
Function , formula , domain
Function , formula , domain
Function , formula , domain is
Function , domain
Composite Functions:
Composition is another method for combining functions.
Definition:
If f and g are functions, the composite function (f “circle” g) is defined by . The domain of consists of the numbers x in the domain of g for which lies in the domain of f.
The definition says that two functions can be composed when the image of the first lies in the domain of the second. To we first find and second find .
Clearly, in general, . That is, composition of functions is not commutative.
Example 6:
If and , find (a) (b) (c) (d)
Solution 6:
a) , domain is
b) , domain is
c) , domain is
d) , domain is or
Even functions and odd functions:
A function f(x) is said to be even if . That is, the function possesses symmetry about the y-axis. Example, .
A function f(x) is said to be odd if . That is, the function possesses symmetry about the origin. Example .
Any function can be expressed as a sum of an even function and an odd function.
A function could be neither even nor odd.
Note that a function like possesses symmetry about the x-axis !!
Piecewise Defined Functions:
Sometimes a function uses different formulas or formulae over different parts of its domain. One such example is the absolute value function:
, when and , when .
Example 7:
The function , when , , when , and , when .
Example 8:
The greatest integer function:
The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted by .
Observe that ; ; ; ; ; ‘ ; .
Example 9:
The least integer function:
The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted by . For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot which charges USD 1 for each hour or part of an hour.
Cheers,
Nalin Pithwa
[contact-form]
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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 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 and . Also, if and only if .
Example 1: Find x and y when .
Solution 1: Equating the first components and then equating the second components, we have:
and
and
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 , .
Thus,
e.g., if and , tnen .
If or , we define .
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: .
So, in general if a cartesian product of p finite sets, viz, is given by
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 is called relation from A to B, and is denoted by capital letters P, Q and R. If R is a relation and then it is denoted by .
y is called image of x under R and x is called pre-image of y under R.
Let and .
Let R be a relation such that implies . We list the elements of R.
Solution: Here and so that Note this is the relation R from A to B, that is, it is a subset of A x B.
Check: Is a relation from B to A defined by x<y, with and — is this relation *same* as R from A to B? Ans: Let us list all the elements of R^{‘} explicitly: . Well, we can surely compare the two sets R and — the elements “look” different certainly. Even if they “look” same in terms of numbers, the two sets and 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 , then domain (R) is .
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 , then range (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 , and let and let Then is a one-one relation. Here, domain of and range of is .
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 . Then, is many-one relation from A to B. (please draw arrow diagram). Note also that domain of and range of .
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 and . Consider the relation . So, clearly range is and . Thus, 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 and set . Let . So, clearly range of . Range of is co-domain of B. Thus, is a relation from A ONTO B.
Binary Relation on a set A:
Let A be a non-empty set then every subset of is a binary relation on set A.
Illustrative Examples:
E.g.1: Let and let . Now, if we have a set then we observe that , and hence, R is a binary relation on A.
E.g.2: Let N be the set of natural numbers and . Since , R is a binary relation on N. Clearly, . Also, for the sake of completeness, we state here the following: Domain of R is and Range of R is , codomain of R is N.
Note: (i) Since the null set is considered to be a subset of any set X, so also here, , and hence, is a relation on any set A, and is called the empty or void relation on A. (ii) Since , we say that is a relation on A called the universal relation on A.
Note: Let the cardinality of a (finite) set A be and that of another set B be , then the cardinality of the cartesian product . So, the number of possible subsets of is 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 for all , that is, aRa for all . (ii) Symmetric: If for all (iii) Transitive: If , and , then so also .
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 defined by . Prove that R is an equivalence relation.
Proof: Given that . (i) Let then , hence, , so relation R is reflexive. (ii) Now, note that , that is, is an integer . That is, we have proved and so relation R is symmetric also. (iii) Now, let , and , which in turn implies that and so it (as integers are closed under addition) which in turn . Thus, and implies also, Hence, given relation R is transitive also. Hence, R is also an equivalence relation on .
Illustrative examples continued:
E.g.: If , 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: and so the solution is .
E.g.: If , list the set .
Solution:
E.g.: If and , find , and , check if cartesian product is a commutative operation, that is, check if .
Solution: whereas so since so cartesian product is not a commutative set operation.
E.g.: If two sets A and B are such that their cartesian product is , 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 and .
E.g.: A and B are two sets given in such a way that contains 6 elements. If three elements of are , find its remaining elements.
Solution: We can first observe that 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 and and so we have found the sets A and B completely.
E.g.: Express the set as a set of ordered pairs.
Solution: We have and so
Hence, the given set is
E.g.: Let and . Show that is a relation from A to B. Find the domain, co-domain and range.
Solution: Here, . Clearly, . 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) = and co-domain of R is set B itself; and Range of R is .
E.g.: Let and . Let R be a relation from A to B such that if . List all the elements of R. Find the domain, codomain and range of R. (as homework quiz, draw its arrow diagram);
Solution: Let and . So, we get R as . , , and .
E.g. Let . Define a binary relation on A such that . Find the domain, codomain and range of R.
Solution: By definition, . Here, we get . So we get , ,
Tutorial problems:
More later,
Nalin Pithwa
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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 . And, we read it as “x belongs to A.” If x is not an element of a set A, we write , and read as ‘x does not belong to A.’e.g., 1 is a “whole” number but not a “natural” number.
Hence, , where W is the set of whole numbers and , 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?
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:
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 :
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, .
Superset: If , then B is called the superset of set A. Symbolically:
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 , but then A is called a proper subset of B and we write .
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:
Types of Sets:
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 and . 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, and . Thus, the sets are equal sets and we denote it by .
How to prove that two sets are equal?
Let us say we are given the task to prove that , where A and B are non-empty sets. The following are the steps of the proof : (i) TPT: , that is, choose any arbitrary element and show that also holds true. (ii) TPT: , that is, choose any arbitrary element , and show that also . (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, ? The standard way is similar to above approach: (i) TPT: (ii) TPT: . Both (i) and (ii) together imply that .
4. Equivalent sets: Two finite sets A and B are said to be equivalent if . Equal sets are always equivalent but equivalent sets need not be equal. For example, let and . Then, , so A and B are equivalent. Clearly, . 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 N the set of integers Z are subsets of set of real numbers R. Thus, for this discussion R is a universal set. In general, universal set is denoted by U 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:
Let us now present some (easily provable/verifiable) properties of sets:
Similarly, some easily verifiable properties of set intersection are:
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) . In words, it can be captured beautifully: the complement of union is intersection of complements.
ii) . 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 .
Inclusion Exclusion Principle:
For two sets A and B, given a universal set U: .
For three sets A, B and C, given a universal set U: .
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 , then . Symbol: .
Homework Tutorial I:
5i)
5ii)
5iii)
5iv)
5v)
6. If A and B are subsets of the universal set is X, , , , , find (i) (ii) (iii) (iv)
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 write down the power set of A.
12) Write the following intervals in Set Builder Form: (a) (b) (c) (d)
13) Using Venn Diagrams, represent (a) (b) (c) (d)
Regards,
Nalin Pithwa.
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