PROPERTIES OF SET OPERATIONS

When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out.

The following are the important properties of set operations.   

(i) Commutative Property : 

(a)  A u B  =  B u A

(Set union is commutative)

(b)  A n B  =  B n A

(Set intersection is commutative)

(ii) Associative Property :

(a)  A u (B u C)  =  (A u B) u C

(Set union is associative)

(b)  A n (B n C)  =  (A n B) n C

(Set intersection is associative)

(iii) Distributive Property :

(a)  A n (B u C)  =  (A n B) u (A n C)

(Intersection distributes over union)

(a)  A u (B n C)  =  (A u B) n (A u C)

(Union distributes over intersection)

Problem 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) Set union is commutative. Also verify it by using Venn diagram. 

(ii) Set intersection is commutative. Also verify it by using Venn diagram.

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

By Venn diagram, we have 

From the above two Venn diagrams, it is clear that 

A u B  =  B u A

Hence, it is verified that set union is commutative.

(ii) Let us verify that union is commutative. 

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

By Venn diagram, we have 

From the above two Venn diagrams, it is clear that 

A n B  =  B n A

Hence, it is verified that set intersection is commutative.

Problem 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

By Venn diagram, we have 

From the above Venn diagrams (2) and (4), it is clear that 

A u (B u C)  =  (A u B) u C

Hence, it is verified that set union is associative.

Problem 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

By Venn diagram, we have 

From the above Venn diagrams (2) and (4), it is clear that 

A n (B n C)  =  (A n B) n C

Hence, it is verified that set intersection is associative.

Problem 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)

By Venn diagram, we have 

From the above Venn diagrams (2) and (5), it is clear that 

A u (B n C)  =  (A u B) n (A u C)

Hence, it is verified that union distributes over intersection.

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