Sets

Set

Set is a collection of objects (elements).

A = { 1, 2, 3, 4, 5}. This set has 5 elements.

The above set can also be written as A = {x: x is natural number less than 6}

or A = {x: x ∊ N and x < 6}

or A = {x: x ∊ N and x ≤ 5}

Set of even numbers less than 10 – {2, 4, 6, 8}

Set of prime numbers less than 20 – {2, 3, 5, 7, 11, 13, 17, 19}

Set of English months that starts with the letter J – {January, June, July}

Equal Sets

Two sets are said to be equal if they have same elements.

A = { 1,2,3,4,5} B = { 1,5,3,4,2} C = { 1,3,5,7,9}

A and B are equal sets, A = B.

A and C are not equal sets, A ≠ C.

Null Set

Null set has no elements.

It is denoted by ∅ or { }.

Union of Sets

A = { 1, 2, 3, 4, 5 } B = { 2, 4, 5, 7, 8 }

Union of two sets is the set of all elements that appear in either set.

A U B = { 1, 2, 3, 4, 5, 7, 8 }

If an element appears in both sets, write it only once.

Intersection of Sets

A = { 1, 2, 3, 4, 5 } B = { 2, 4, 5, 7, 8 }

Intersection of two sets is the set of all elements that appear in both sets.

A ∩ B = { 2, 4, 5 }

Subset

Set A is a subset of B, if all elements of A are in B.

A = { 2, 4, 5 }

B = { 2, 4, 5, 7, 8 }

We write as A ⊂ B, A is proper subset of B.

We write A ⊆ A, A is a subset of A.

Subsets of A = { 2, 4, 5} are

{2, 4, 5}, {2, 4}, {4, 5}, {2, 5}, {2}, {4}, {5}, { }

If a set has 3 elements, it will have 2³ = 8 subsets.

If a set has n elements, it will have 2ⁿ subsets.

Universal Set and Complement of a Set

Universal Set is the set of all elements in consideration.

Universal Set is denoted by ε.

ε = { 1, 2, 3, ….,10 }

Complement of set A, denoted by A’ is the set of all elements of Universal Set which are not in A.

If A = {2, 4, 5, 7, 8} then A’ = {1, 3, 6, 9, 10}

Venn Diagram