Basic Set Theory

What is a SET?

A Set is a collection of objects or elements. Sets will be named with Capital alphabets and their elements will be denoted by small alphabets.

Examples:

A = {a, b, c, d}, A is a set of 4 elements a, b, c, d.

B = {1, 2, 3, …, 10}, B is the set of 10 elements from 1 to 10.

N = {1, 2, 3, …}, N is the set of natural numbers whose elements are the numbers 1, 2, 3, …

What are Elements of SET?

The members of a set are called its elements.

In the set A = {a, b, c, d}, a is an element of A, b is an element of A and c, d are elements of A which are symbolically written as,

a∈A, b∈A and c, d∈A

Read as, the element ‘a’ belongs to the set A, the element ‘b’ belongs to the set A and the elements ‘c’ and ‘d’ belongs to the set A.

Way of defining a SET:

A set can be defined in 3 ways.

Listing all elements of a set with in set braces { } such as C = {22, 24, 26, …}
Stating a property of the set such as C = {x| x is an even number and x > 20}, read as C is the set of all x such that x is an even number which are greater than 20.
Set of rules defining the elements of the set such as

22 ∈ C,
If x∈ C then x + 2∈ C,
Nothing else belong to C

Here the set C defined by method 2 and method 3 are the same.

Identical Sets:

Two sets are said to be identical if both sets have same collection of elements. For example, if A = {a, b, c, d} and B = {a, b, c, d} then A and B are said to be identical and will be written as, A = B.

Cardinality of a Set:

The number of elements in a set A is said to be its cardinality and will be written as |A|. For A = {a, b, c, d}, |A| = 4.

Subset of a Set:

A set B is said to be a Subset of a set A, if all elements of the set B are also the elements of A and is written as

B⊆A

and read as B is a subset of A.

Proper subset of a Set:

If B is a subset of A and B is not identical to A then B is said to be a Proper subset of A and written as,

If B⊆A and B≠A then B⊂A

Empty Set:

A set with no element is said to be an Empty set. Empty set is the subset of every set and is denoted by ∅.

Power Set:

The set of all subsets of a set A is said to be its power set and is denoted as P(A).

If A = {a, b, c, d} then its power set is written as,

P(A) = { ∅, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d} }

Cardinality of the power set of A is ,

Cardinality of P(A) = 2^{Cardinality of A}
i.e., |P(A)|= 2^|A| = 2⁴=16

Universal Set:

The set to which all our sets are subsets of, is said to be the universal set. It is denoted by the letter U.

Example:

For A = {a, b, c, d} the universal set is the set of alphabets U = {a, b, c, …, z}

For B= {1, 2, 3, 4, 5, 6} the universal set is the set of natural numbers U = {1, 2, 3, …}