Types of sets

The Empty set

A set which does not contain any element is called the empty set or the null set or the  void set.

Example : D = {x : x^2=9 , x is even  }

Here D is the empty set, because the equation x^2 = 9  is not satisfied by any even value of  x.

Note : The empty set is denoted by the symbol φ or { }.

Singleton Set

A set which contains only one element is called a singleton set.

Example : A = {x : x is neither prime nor composite}

It is a singleton set containing one element, i.e., 1.

Example : B = {x : x is a whole number, x < 1}

This set contains only one element 0 and is a singleton set.

Example : Let A = {x : x ∈ N and x² = 4}

Here A is a singleton set because there is only one element 2 whose square is 4.

Here A is a singleton set because there is only one element 2 whose square is 4.

Finite Set

A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.

Example : The set of all colours in the rainbow.

Example : N = {x : x ∈ N, x < 7}

Example : P = {2, 3, 5, 7, 11, 13, 17, ...... 97}

Infinite Set

The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.

Example : Set of all points in a plane

Example :  A = {x : x ∈ N, x > 1}

Example : Set of all prime numbers

Example :  B = {x : x ∈ W, x = 2n}

Note: All infinite sets cannot be expressed in roster form.

Example : The set of real numbers since the elements of this set do not follow any particular pattern.

Equal Sets

Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.

Example :  

A = {p, q, r, s}

B = {p, s, r, q}

Therefore, A = B

Equivalent Set

Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.

Example :  

A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3

Therefore, A ↔ B

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.

Example :  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]

Example :  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.

Example :  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.

Subsets

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.

Example :  Let A = {2, 4, 6}   B = {6, 4, 8, 2}

Solution : 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:

1.      If A ⊂ B and B ⊂ A, then A = B, i.e., they are equal sets.

2.      Every set is a subset of itself.

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

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

Example :  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

Example :  Let A = {1, 2, 3, 4}

B = {4, 5, 6, 7}

Here A ⊄ B and also B ⊄ C

[⊄ denotes ‘not a subset of’]

Proper Subsets

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.

Example :  A = {1, 2, 3, 4}     B = {1, 2, 3, 4, 5}

Solution : Here n(A) = 4,   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.

Example :  A = {p, q, r}     B = {p, q, r, s, t}

Solution : 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.

Superset

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’

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 Superset

A proper superset of a set A is a superset of A that is not equal to A.

In other words, if B is a proper superset of A, then all elements of A are in B but B contains at least one element that is not in A.

Example : if A={1,3,5} then B={1,3,4,5} is a proper superset of A. The set C={1,3,5} is a superset of A, but it is not a proper superset of A since C=A. The set D={1,3,7} is not even a superset of A, since D does not contain the element 5.

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.

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.

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