LAWS OF SET OPERATIONS

Commutative Law : 

A u B  =  B u A

A n B  =  B n A

Associative Law :

A u (B u C)  =  (A u B) u C

A n (B n C)  =  (A n B) n C

Distributive Law :

A n (B u C)  =  (A n B) u (A n C)

A u (B n C)  =  (A u B) n (A u C)

Identity Law :

AU∅  =  A

A n U  =  A

Complement Law :

AUA'  =  U

A n A'  =  

Double Complement Law :

(A')'  =  A

Idempotent Law :

AUA  =  A

A n A  =  A

Universal Bound Law :

AuU  =  U

An  =  

De morgan's Law

(AuB)'  =  A'nB'

(AnB)'  =  A'uB'

Absorption Law :

Au(AnB)  =  A

An(AUB)  =  A

Examples On Laws of Set Operation

Example 1 :

For the given sets

A  =  {-10, 0, 1, 9, 2, 4, 5}

B  =  {-1, -2, 5, 6, 2, 3, 4},

Verify the following :

(i)  AuB  =  BuA

(ii)  AnB  =  BnA

Solution : 

(i) Let us verify that union is commutative. 

A u B  =  {-10, 0, 1, 9, 2, 4, 5} u {-1, -2, 5, 6, 2, 3, 4}

A u B  =  {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ---------(1)

B u A  =  {-1, -2, 5, 6, 2, 3, 4 } u {-10, 0, 1, 9, 2, 4, 5} 

B u A  =  {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ---------(2)

From (1) and (2), we have 

A u B  =  B u A

(ii)

A n B  =  {-10, 0, 1, 9, 2, 4, 5} n {-1, -2, 5, 6, 2, 3, 4}

A n B  =  {2, 4, 5} ---------(1)

B n A  =  {-1, -2, 5, 6, 2, 3, 4 } u {-10, 0, 1, 9, 2, 4, 5} 

B n A  =  {2, 4, 5} ---------(2)

From (1) and (2), we have 

A n B  =  B n A

Example 2 :

For the given sets

A  =  {1, 2, 3, 4, 5}, B  =  {3, 4, 5, 6}, C  =  {5, 6, 7, 8}

verify that

A u (B u C )  =  (A u B) u C

Also verify it by using Venn diagram. 

Solution : 

Let us verify that set union is associative. 

B u C  =  {3, 4, 5, 6} u {5, 6, 7, 8}

B u C  =  {3, 4, 5, 6, 7, 8}

A u (B u C)  =  {1, 2, 3, 4, 5} u {3, 4, 5, 6, 7, 8} 

A u (B u C)  =  {1, 2, 3, 4, 5, 6, 7, 8} ---------(1)

A u B  =  {1, 2, 3, 4, 5} u {3, 4, 5, 6}

A u B  =  {1, 2, 3, 4, 5, 6}

(A u B) u C  =  {1, 2, 3, 4, 5, 6} u {5, 6, 7, 8} 

(A u B) u C  =  {1, 2, 3, 4, 5, 6, 7, 8} ---------(2)

From (1) and (2), we have 

A u (B u C)  =  (A u B) u C

Example 3 :

For the given sets

A  =  {a, b, c, d}, B  =  {a, c, e}, C  =  {a, e}

verify that

A n (B n C)  =  (A n B) n C

Also verify it by using Venn diagram. 

Solution :

Let us verify that set intersection is associative. 

B n C  =  {a, c, e} u {a, e}

B n C  =  {a, e}

A n (B n C)  =  {a, b, c, d} n {a, e} 

A n (B n C)  =  {a} ---------(1)

A n B  =  {a, b, c, d} u {a, c, e}

A n B  =  {a, c}

(A n B) n C  =  {a, c} n {a, e} 

(A n B) n C  =  {a} ---------(2)

From (1) and (2), we have 

A n (B n C)  =  (A n B) n C

Example 4 :

For the given sets

A  =  {0, 1, 2, 3, 4}, B  =  {1, -2, 3, 4, 5, 6}, C  =  {2, 4, 6, 7}

verify that

A u (B n C )  =  (A u B) n (A u C)

Also verify it by using Venn diagram. 

Solution : 

Let us verify that union distributes over intersection. 

B n C  =  {1, -2, 3, 4, 5, 6} n {2, 4, 6, 7}

B n C  =  {4, 6}

A u (B n C)  =  {0, 1, 2, 3, 4} u {4, 6} 

A u (B n C)  =  {0, 1, 2, 3, 4, 6} -----(1)

A u B  =  {0, 1, 2, 3, 4} u {1, -2, 3, 4, 5, 6}

A u B  =  {-2, 0, 1, 2, 3, 4, 5, 6}

A u C  =  {0, 1, 2, 3, 4 } u {2, 4, 6, 7}

A u C  =  {0, 1, 2, 3, 4, 6, 7}

(A u B)n(A u C) = {-2, 0, 1, 2, 3, 4, 5, 6}n{0, 1, 2, 3, 4, 6, 7}

(A u B) n (A u C)  =  {0, 1, 2, 3, 4, 6} ---------(1)

From (1) and (2), we have 

A u (B n C)  =  (A u B) n (A u C)

Example 5 :

Let U  =  {1, 2, 3, 4, ......10}, A  =  {5, 6, 7, 9}, find A'. 

Solution :

A'  =  U \ A

A'  =  {1, 2, 3, 4, ......10} \ {5, 6, 7, 9}

A'  =  {1, 2, 3, 4, 8, 10}

Example 6 :

Let A  =  {1, 3, 5, 6}, B  =  {0, 5, 6, 7}, find A \ B. 

Solution :

A \ B  =  {1, 3, 5, 6} \ {0, 5, 6, 7}

A \ B  =  {1, 3}

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