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Basics of Set Theory-Part 3( Subsets )

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 Ì X. The symbol Ì

stands for ‘is a subset of’ or ‘is contained in’.

A set A is a subset of a set B if every element of A is also an element of  B.

In other words, A Ì B if whenever a Î A, then a Î B. It is often convenient to

use the symbol Þ which means implies. Using this symbol, we can write the definition

of subset as follows:

A Ì B if a Î A Þ a Î B   

above statement is read 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 Ë 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 Ì A. In this case, A and B are the same sets so that we have A Ì B and B Ì A Û A = B, where Û 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 Ì A. Since the empty set f has no elements, we agree to say that f is a subset of

every set. We now consider some examples :

 

(i) The set Q of rational numbers is a subset of the set R of real numbes, and

we write Q Ì R.

(ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56,

then B is a subset of A and we write B Ì A.

(iii) Let A = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then

A Ì B and B Ì 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 Ì B and B Ë A  then A is called a proper subset

of  B and B is called superset of A. For example,

A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.

If a set A has only one element, we call it a singleton set. Thus,{ a } is a

singleton set.

Example 9   Consider the sets

f, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.

Insert the symbol Ì or Ë between each of the following pair of sets:

(i) ö f. . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

Solution (i) öf Ì B as f is a subset of every set.

(ii) A Ë B as 3 Î A and 3 Ï B

(iii) A Ì C as 1, 3 Î A also belongs to C

(iv) B Ì C as each element of B is also an element of C.

Example 10   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 11   Let A, B and C be three sets. If A Î B and B Ì C, is it true that

A Ì C?. If not, give an example.

Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here AÎ B as A = {1}

and B Ì C. But A Ë C as 1 Î A and 1 Ï C.

Note that an element of a set can never be a subset of itself.

 

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