*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.