Subset

If A and B are
two sets, and every element of set A is also an element of set B, then A
is called a subset of B and we write it as A ⊆ B or B ⊇ A

The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’ 

• Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B. 

• Empty set is a subset of every set. 

• Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’. 

• A ⊆ B means A is a subset of B or A is contained in B. 

• B ⊆ A means B contains A. 

For example;

1. Let A = {2, 4, 6}

B = {6, 4, 8, 2}

Here A is a subset of B

Since, all the elements of set A are contained in set B.

But B is not the subset of A

Since, all the elements of set B are not contained in set A.

Notes:

If ACB and BCA, then A = B, i.e., they are equal sets.

Every set is a subset of itself.

Null set or ∅ is a subset of every set.

2. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.

3. Let A = {2, 4, 6}

B = {x : x is an even natural number less than 8}

Here A ⊂ B and B ⊂ A.

Hence, we can say A = B

4. Let A = {1, 2, 3, 4}

B = {4, 5, 6, 7}

Here A ⊄ B and also B ⊄ C

[⊄ denotes ‘not a subset of’]

Super Set:

Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.

Symbol ⊇ is used to denote ‘is a super set of’

For example;

A = {a, e, i, o, u}

B = {a, b, c, …………., z}

Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A

Proper Subset:

If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B.

For example;

1. A = {1, 2, 3, 4}

Here n(A) = 4

B = {1, 2, 3, 4, 5}

Here n(B) = 5

We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A.

So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B

Notes:

No set is a proper subset of itself.

Null set or ∅ is a proper subset of every set.

2. A = {p, q, r}

B = {p, q, r, s, t}

Here A is a proper subset of B as all the elements of set A are in set B and also A ≠ B.

Notes:

No set is a proper subset of itself.

Empty set is a proper subset of every set.

Power Set:

The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

For example;

If A = {p, q} then all the subsets of A will be

P(A) = {∅, {p}, {q}, {p, q}}

Number of elements of P(A) = n[P(A)] = 4 = 22

In general, n[P(A)] = 2m where m is the number of elements in set A.

Universal Set

A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is ∪ or ξ.

For example;

1. If A = {1, 2, 3}      B = {2, 3, 4}      C = {3, 5, 7}

then U = {1, 2, 3, 4, 5, 7}

[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]

2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set is a set of all integers.

3. If A = {a, b, c}      B = {d, e}      C = {f, g, h, i}

then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.