## Sets - Operations Of Sets

Class 11 - Maths

**COMPLEMENT OF A SET**

If U be the universal set and A is the subset of U, then the complement of A with respect to U, denoted by A',

A' is defined as

A' = {x : x ∈ U and x ∉ A }

i.e. the complement of A is the set of those elements of U which are not elements of A.

OPERATIONS OF SETS

Main set operations are:

i. Union of Sets

ii. Intersection of Sets

iii. Difference of Sets

**UNION OF TWO SETS**

The union of any two given sets A and B is the set C which consists of all those elements which are either in A or in B.

Symbolically, we write

C = A ∪ B = {x | x ∈A or x ∈B}

Example:

A = {1,2,3,4} and B = {2,4,6,8}

A ∪ B = {1,2,3,4,6,8}

Some properties of the operation of union are:

(i) A ∪ B = B ∪ A

(ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)

(iii) A ∪ φ = A

(iv) A ∪ A = A

(v) U ∪ A = U

**INTERSECTION OF SETS**

The intersection of two sets A and B is denoted by A ∩ B i.e. it is the set which consists of all those elements which belong to both A and B.

Symbolically, we write

A ∩ B = {x : x ∈ A and x ∈ B}.

Example:

A = {2,4,6,8} and B = {6, 8, 10, 12}

A ∩ B = {6, 8}

**DISJOINT SETS**

Two sets A and B are said to be disjoint if, A ∩ B = φ

e.g. A = {1,2,3,4} and B = {6,7,8} then A ∩ B = φ. Its Venn diagram representation is:

Some properties of the operation of intersection

(i) A ∩ B = B ∩ A

(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

(iii) φ ∩ A = φ ; U ∩ A = A

(iv) A ∩ A = A

(v) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(vi) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

**DIFFERENCE OF TWO SETS**

Difference of two sets is denoted by A - B. is defined as set of elements which belong to A but not to B. Symbolically,

A – B = {x : x ∈ A and x ∉ B}

also, B – A = { x : x ∈ B and x ∉A}

Examples:

A = {1, 2, 3, 4, 6, 12} and B = {1, 2, 4, 8, 16}

A - B = {3, 6, 12}

B - A = {8, 16}

**☛See also:**

Special Mathematical Constants

SETS (Unit Test Paper)

SETS (VENN DIAGRAMS)

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