A Note All infinite sets cannot be described in the roster form. For example, the

A Note All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern. Example State which of the following sets are finite or infinite : { x : x ∈ N and ( x – ) ( x – ) = } { x : x ∈ N and x = } (iii) { x : x ∈ N and x – = } (iv) { x : x ∈ N and x is prime} (v) { x : x ∈ N and x is odd} Solution Given set = { , }. Hence, it is finite.

Given set = { }. Hence, it is finite. (iii) Given set = φ . Hence, it is finite.

(iv) The given set is the set of all prime numbers and since set of prime numbers is infinite. Hence the given set is infinite (v) Since there are infinite number of odd numbers, hence, the given set is infinite. . Equal Sets Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal.

Clearly, the two sets have exactly the same elements. Definition Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B. We consider the following examples : Let A = { , , , } and B = { , , , }.

Then A = B. Let A be the set of prime numbers less than and P the set of prime factors of . Then A and P are equal, since , and are the only prime factors of and also these are less than . A Note A set does not change if one or more elements of the set are repeated.

For example, the sets A = { , , } and B = { , , , , } are equal, since each MATHEMATICS element of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set. Example Find the pairs of equal sets, if any, give reasons: A = { }, B = { x : x > and x < }, C = { x : x – = }, D = { x : x = }, E = { x : x is an integral positive root of the equation x – x – = }. Solution Since ∈ A and does not belong to any of the sets B, C, D and E, it follows that, A ≠ B, A ≠ C, A ≠ D, A ≠ E.

Since B = φ but none of the other sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E. Also C = { } but – ∈ D, hence C ≠ D. Since E = { }, C = E.

Further, D = {– , } and E = { }, we find that, D ≠ E. Thus, the only pair of equal sets is C and E. Example Which of the following pairs of sets are equal? Justify your answer.

X, the set of letters in “ALLOY” and B, the set of letters in “LOYAL”. A = { n : n ∈ Z and n ≤ } and B = { x : x ∈ R and x – x + = }. Solution (i) We have, X = {A, L, L, O, Y}, B = {L, O, Y, A, L}. Then X and B are equal sets as repetition of elements in a set do not change a set.

Thus, X = {A, L, O, Y} = B (ii) A = {– , – , , , }, B = { , }. Since ∈ A and ∉ B, A and B are not equal sets. EXERCISE . .

Which of the following are examples of the null set Set of odd natural numbers divisible by Set of even prime numbers (iii) { x : x is a natural numbers, x < and x > } (iv) { y : y is a point common to any two parallel lines} . Which of the following sets are finite or infinite The set of months of a year { , , , . . .} (iii) { , , , .

. . , } (iv) The set of positive integers greater than (v) The set of prime numbers less than . State whether each of the following set is finite or infinite: The set of lines which are parallel to the x -axis The set of letters in the English alphabet (iii) The set of numbers which are multiple of SETS (iv) The set of animals living on the earth (v) The set of circles passing through the origin ( , ) .

In the following, state whether A = B or not: A = { a , b , c , d } B = { d , c , b , a } A = { , , , } B = { , , , } (iii) A = { , , , , } B = { x : x is positive even integer and x ≤ } (iv) A = { x : x is a multiple of }, B = { , , , , , . . . } .

Are the following pair of sets equal ? Give reasons. A = { , }, B = { x : x is solution of x + x + = } A = { x : x is a letter in the word FOLLOW} B = { y : y is a letter in the word WOLF} . From the sets given below, select equal sets : A = { , , , }, B = { , , , }, C = { , , , }, D = { , , , } E = {– , }, F = { , a }, G = { , – }, H = { , } .

Subsets Consider the sets : X = set of all students in your school, Y = set of all students in your class. We note that every element of Y is also an element of X; we say that Y is a subset of X. The fact that Y is subset of X is expressed in symbols as Y ⊂ X. The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.

Definition A set A is said to be a subset of a set B if every element of A is also an element of B. In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “ ⇒ ” which means implies . Using this symbol, we can write the definiton of subset as follows: A ⊂ B if a ∈ A ⇒ a ∈ B We read the above statement as “ A is a subset of B if a is an element of A implies that a is also an element of B ”.

If A is not a subset of B, we write A ⊄ B. We may note that for A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A. If it so happens that every element of B is also in A, then we shall also have B ⊂ A.

In this case, A and B are the same sets so that we have A ⊂ B and B ⊂ A ⇔ A = B, where “ ⇔ ” is a symbol for two way implications, and is usually read as if and only if (briefly written as “iff”). It follows from the above definition that every set A is a subset of itself, i.e., A ⊂ A. Since the empty set φ has no elements, we agree to say that φ is a subset of every set . We now consider some examples : MATHEMATICS The set Q of rational numbers is a subset of the set R of real numbes, and we write Q ⊂ R.

If A is the set of all divisors of and B the set of all prime divisors of , then B is a subset of A and we write B ⊂ A. (iii) Let A = { , , } and B = { x : x is an odd natural number less than }. Then A ⊂ B and B ⊂ A and hence A = B. (iv) Let A = { a, e, i, o, u } and B = { a, b, c, d }.

Then A is not a subset of B, also B is not a subset of A. Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A. For example, A = { , , } is a proper subset of B = { , , , }.

If a set A has only one element, we call it a singleton set . Thus,{ a } is a singleton set. Example Consider the sets φ , A = { , }, B = { , , }, C = { , , , , }. Insert the symbol ⊂ or ⊄ between each of the following pair of sets: (i) φ .

C (iv) B . . . C Solution φ ⊂ B as φ is a subset of every set.

A ⊄ B as ∈ A and ∉ B (iii) A ⊂ C as , ∈ A also belongs to C (iv) B ⊂ C as each element of B is also an element of C. Example Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B ? No.

(Why?). Is B a subset of A? No. (Why?) Example Let A, B and C be three sets.

If A ∈ B and B ⊂ C, is it true that A ⊂ C?. If not, give an example. Solution No. Let A = { }, B = {{ }, } and C = {{ }, , }.

Here A ∈ B as A = { } and B ⊂ C. But A ⊄ C as ∈ A and ∉ C. Note that an element of a set can never be a subset of itself. .

. Subsets of set of real numbers As noted in Section . , there are many important subsets of R . We give below the names of some of these subsets.

The set of natural numbers N = { , , , , , . . .} The set of integers Z = {. .

., – , – , – , , , , , . . .} The set of rational numbers Q = { x : x = p q , p, q ∈ Z and q ≠ } SETS which is read “ Q is the set of all numbers x such that x equals the quotient p q , where p and q are integers and q is not zero”. Members of Q include – (which can be expressed as – ) , , (which can be expressed as ) and – The set of irrational numbers, denoted by T , is composed of all other real numbers.

Thus T = { x : x ∈ R and x ∉ Q }, i.e., all real numbers that are not rational. Members of T include , and π . Some of the obvious relations among these subsets are: N ⊂ Z ⊂ Q , Q ⊂ R , T ⊂ R , N ⊄ T . .

. Intervals as subsets of R Let a, b ∈ R and a < b. Then the set of real numbers { y : a < y < b } is called an open interval and is denoted by ( a , b ) . All the points between a and b belong to the open interval ( a, b ) but a, b themselves do not belong to this interval.

The interval which contains the end points also is called closed interval and is denoted by [ a, b ]. Thus [ a, b ] = { x : a ≤ x ≤ b } We can also have intervals closed at one end and open at the other, i.e., [ a, b ) = { x : a ≤ x < b } is an open interval from a to b, including a but excluding b. ( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a. These notations provide an alternative way of designating the subsets of set of real numbers.

For example , if A = (– , ) and B = [– , ], then A ⊂ B. The set [ , ∞ ) defines the set of non-negative real numbers, while set ( – ∞ , ) defines the set of negative real numbers. The set ( – ∞ , ∞ ) describes the set of real numbers in relation to a line extending from – ∞ to ∞ . On real number line, various types of intervals described above as subsets of R , are shown in the Fig .

. Here, we note that an interval contains infinitely many points. For example, the set { x : x ∈ R , – < x ≤ }, written in set-builder form, can be written in the form of interval as (– , ] and the interval [– , ) can be written in set- builder form as { x : – ≤ x < }. Fig .

MATHEMATICS The number ( b – a ) is called the length of any of the intervals ( a, b ), [ a, b ], [ a, b ) or ( a, b ] . . Power Set Consider the set { , }. Let us write down all the subsets of the set { , }.

We know that φ is a subset of every set . So, φ is a subset of { , }. We see that { } and { }are also subsets of { , }. Also, we know that every set is a subset of itself.

So, { , } is a subset of { , }. Thus, the set { , } has, in all, four subsets, viz. φ , { }, { } and { , }. The set of all these subsets is called the power set of { , }.

Definition The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set. Thus, as in above, if A = { , }, then P( A ) = { φ ,{ }, { }, { , }} Also, note that n [ P (A) ] = = In general, if A is a set with n (A) = m , then it can be shown that n [ P(A)] = m .

. Universal Set Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “ Universal Set ”.

The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. For another example, in human population studies, the universal set consists of all the people in the world. EXERCISE .

. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces : { , , } . . .

{ , , , , } (ii) { a , b , c } . . . { b , c , d } (iii) { x : x is a student of Class XI of your school}.

. .{ x : x student of your school} (iv) { x : x is a circle in the plane} . . .{ x : x is a circle in the same plane with radius unit} (v) { x : x is a triangle in a plane} .

. . { x : x is a rectangle in the plane} (vi) { x : x is an equilateral triangle in a plane} . .

. { x : x is a triangle in the same plane} (vii) { x : x is an even natural number} . . .

{ x : x is an integer} SETS Fig . . Examine whether the following statements are true or false: { a , b } ⊄ { b , c, a } { a , e } ⊂ { x : x is a vowel in the English alphabet} (iii) { , , } ⊂ { , , } (iv) { a } ⊂ { a , b, c } (v) { a } ∈ { a , b, c } (vi) { x : x is an even natural number less than } ⊂ { x : x is a natural number which divides } . Let A = { , , { , }, }.

Which of the following statements are incorrect and why? { , } ⊂ A { , } ∈ A (iii) {{ , }} ⊂ A (iv) ∈ A (v) ⊂ A (vi) { , , } ⊂ A (vii) { , , } ∈ A (viii) { , , } ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) { φ } ⊂ A . Write down all the subsets of the following sets { a } { a , b } (iii) { , , } (iv) φ . How many elements has P(A), if A = φ ?

. Write the following as intervals : { x : x ∈ R, – < x ≤ } { x : x ∈ R, – < x < – } (iii) { x : x ∈ R, ≤ x < } (iv) { x : x ∈ R, ≤ x ≤ } . Write the following intervals in set-builder form : (– , ) [ , ] (iii) ( , ] (iv) [– , ) . What universal set(s) would you propose for each of the following : The set of right triangles.

The set of isosceles triangles. . Given the sets A = { , , }, B = { , , } and C = { , , , , }, which of the following may be considered as universal set (s) for all the three sets A, B and C { , , , , , , } φ (iii) { , , , , , , , , , , } (iv) { , , , , , , , } . Venn Diagrams Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams .

Venn diagrams are named after the English logician, John Venn ( - ). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles. In Venn diagrams, the elements of the sets are written in their respective circles (Figs .

and . ) MATHEMATICS Fig . Illustration In Fig . , U = { , , , ..., } is the universal set of which A = { , , , , } is a subset.

Illustration In Fig . , U = { , , , ..., } is the universal set of which A = { , , , , } and B = { , } are subsets, and also B ⊂ A. The reader will see an extensive use of the Venn diagrams when we discuss the union, intersection and difference of sets. .

Operations on Sets In earlier classes, we have learnt how to perform the operations of addition, subtraction, multiplication and division on numbers. Each one of these operations was performed on a pair of numbers to get another number. For example, when we perform the operation of addition on the pair of numbers and , we get the number . Again, performing the operation of multiplication on the pair of numbers and , we get .

Similarly, there are some operations which when performed on two sets give rise to another set. We will now define certain operations on sets and examine their properties. Henceforth, we will refer all our sets as subsets of some universal set. .

. Union of sets Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘ ∪ ’ is used to denote the union .

Symbolically, we write A ∪ B and usually read as ‘ A union B ’ . Example Let A = { , , , } and B = { , , , }. Find A ∪ B. Solution We have A ∪ B = { , , , , , } Note that the common elements and have been taken only once while writing A ∪ B.

Example Let A = { a, e, i, o, u } and B = { a, i, u }. Show that A ∪ B = A Solution We have, A ∪ B = { a, e, i, o, u } = A. This example illustrates that union of sets A and its subset B is the set A itself, i.e., if B ⊂ A, then A ∪ B = A. Example Let X = {Ram, Geeta, Akbar} be the set of students of Class XI, who are in school hockey team.

Let Y = {Geeta, David, Ashok} be the set of students from Class XI who are in the school football team. Find X ∪ Y and interpret the set. Solution We have, X ∪ Y = {Ram, Geeta, Akbar, David, Ashok}. This is the set of students from Class XI who are in the hockey team or the football team or both.

SETS Thus, we can define the union of two sets as follows: Definition The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both). In symbols, we write. A ∪ B = { x : x ∈ A or x ∈ B } The union of two sets can be represented by a Venn diagram as shown in Fig . .

The shaded portion in Fig . represents A ∪ B. Some Properties of the Operation of Union A ∪ B = B ∪ A (Commutative law) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law ) (iii) A ∪ φ = A (Law of identity element, φ is the identity of ∪ ) (iv) A ∪ A = A (Idempotent law) (v) U ∪ A = U (Law of U) . .