If A = - 1 2 , 0 , 1 4 , 3 4 , 2 , B = 0 , 1 4 , 3 4 , 2 , 5 2 and C = - 1 2 , 1 4 , 1 , 2 , 5 2 , then verify A ∩ B ∩ C = A ∩ B ∩ C .

Given, A = - 1 2 , 0 , 1 4 , 3 4 , 2 , B = 0 , 1 4 , 3 4 , 2 , 5 2 and C = - 1 2 , 1 4 , 1 , 2 , 5 2 To verify: A ∩ B ∩ C = A ∩ B ∩ C . We know that, The intersection of sets A a n d B is the set of all elements which are common to both A a n d B . Now, B ∩ C = 0 , 1 4 , 3 4 , 2 , 5 2 ∩ - 1 2 , 1 4 , 1 , 2 , 5 2 B ∩ C = 1 4 , 2 , 5 2 A ∩ B ∩ C = - 1 2 , 0 , 1 4 , 3 4 , 2 ∩ 1 4 , 2 , 5 2 A ∩ B ∩ C = 1 4 , 2 . . . . 1 Then, A ∩ B = - 1 2 , 0 , 1 4 , 3 4 , 2 ∩ 0 , 1 4 , 3 4 , 2 , 5 2 A ∩ B = 0 , 1 4 , 3 4 , 2 A ∩ B ∩ C = 0 , 1 4 , 3 4 , 2 ∩ - 1 2 , 1 4 , 1 , 2 , 5 2 A ∩ B ∩ C = 1 4 , 2 . . . . 2 From 1 and 2 , A ∩ B ∩ C = A ∩ B ∩ C Hence, verified

If A = 0 , 2 , 4 , 6 , 8 , B = { x : x is a prime number and x < 11 } and C = { x : x ∈ N and 5 ≤ x < 9 } then verify A ∪ B ∩ C = A ∪ B ∩ A ∪ C .

Given, A = 0 , 2 , 4 , 6 , 8 , B = { x : x is a prime number and x < 11 } and C = { x : x ∈ N and 5 ≤ x < 9 } To verify: A ∪ B ∩ C = A ∪ B ∩ A ∪ C . A = 0 , 2 , 4 , 6 , 8 , B = 2 , 3 , 5 , 7 and C = 5 , 6 , 7 , 8 We know that, For any three sets A , B a n d C , A ∪ B ∩ C = A ∪ B ∩ A ∪ C is called distributive property of union over intersection. The intersection of sets A a n d B is the set of all elements which are common to both A a n d B . First, we find B ∩ C = 2 , 3 , 5 , 7 ∩ 5 , 6 , 7 , 8 B ∩ C = 5 , 7 A ∪ B ∩ C = 0 , 2 , 4 , 5 , 6 , 7 , 8 . . . . 1 The union of two sets A a n d B is the set of all elements which are either in A or in B or in both. A ∪ B = 0 , 2 , 4 , 6 , 8 ∪ 2 , 3 , 5 , 7 A ∪ B = 0 , 2 , 3 , 4 , 5 , 6 , 7 , 8 A ∪ C = 0 , 2 , 4 , 6 , 8 ∪ 5 , 6 , 7 , 8 A ∪ C = 0 , 2 , 4 , 5 , 6 , 7 , 8 Then, A ∪ B ∩ A ∪ C = 0 , 2 , 4 , 5 , 6 , 7 , 8 . . … 2 From 1 and 2 , it is verified that A ∪ B ∩ C = A ∪ B ∩ A ∪ C .

Verify A ∩ B ∪ C = A ∩ B ∪ A ∩ C using Venn diagrams.

Given to verify: A ∩ B ∪ C = A ∩ B ∪ A ∩ C using Venn diagram We know that, Venn diagram are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way For any three sets A , B a n d C , A ∩ B ∪ C = A ∩ B ∪ A ∩ C is called distributive property of intersection over union. The given property can verify using Venn diagram as shown below: ......... I ....... I I From I a n d I I A ∩ B ∪ C = A ∩ B ∪ A ∩ C is verified Let us consider an example: A = 0 , 2 , 4 , 6 , 8 , B = 2 , 3 , 5 , 7 , C = 5 , 6 , 7 , 8 The union of sets A a n d B is the set of all elements which are either in A or B or both. First, we find B ∪ C = 2 , 3 , 5 , 7 ∪ 5 , 6 , 7 , 8 B ∪ C = 2 , 3 , 5 , 6 , 7 , 8 A ∩ B ∪ C = 2 , 6 , 8 . . . . 1 The intersection of two sets A a n d B is the set of all elements which are common to both A a n d B A ∩ B = 2 A ∩ C = 6 , 8 Then, A ∩ B ∪ A ∩ C = 2 , 6 , 8 . . … 2 From 1 and 2 , it is verified that A ∩ B ∪ C = A ∩ B ∪ A ∩ C .

Verify A - B ∪ C = A - B ∩ A - C using Venn diagrams.

Given to verify: A - B ∪ C = A - B ∩ A - C using Venn diagram We know that, Venn diagram are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way For any three sets A , B a n d C , A - B ∪ C = A - B ∩ A - C is called De Morgan's law for set difference. The given law can verify using Venn diagram as shown below: ......... I ....... I I From I a n d I I A - B ∪ C = A - B ∩ A - C is verified Let us consider an example: A = 0 , 2 , 4 , 6 , 8 , B = 2 , 3 , 5 , 7 , C = 5 , 6 , 7 , 8 The union of sets A a n d B is the set of all elements which are either in A or B or both. First, we find B ∪ C = 2 , 3 , 5 , 7 ∪ 5 , 6 , 7 , 8 B ∪ C = 2 , 3 , 5 , 6 , 7 , 8 A - B ∪ C = 0 , 4 . . . . 1 The intersection of two sets A a n d B is the set of all elements which are common to both A a n d B A - B = 0 , 4 , 6 , 8 A - C = 0 , 2 , 4 Then, A - B ∩ A - C = 0 , 4 . . … 2 From 1 and 2 , it is verified that A - B ∪ C = A - B ∩ A - C .

If P = { x : x ∈ W and 0 < x < 10 } , Q = { x : x = 2 n + 1 , n ∈ W and n < 5 } and R = 2 , 3 , 5 , 7 , 11 , 13 , then verify P - Q ∩ R = P - Q ∪ P - R .

Given, P = { x : x ∈ W and 0 < x < 10 } , Q = { x : x = 2 n + 1 , n ∈ W and n < 5 } and R = 2 , 3 , 5 , 7 , 11 , 13 To verify: P - Q ∩ R = P - Q ∪ P - R . The roster form of sets P , Q and R are P = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , Q = 1 , 3 , 5 , 7 , 9 and R = 2 , 3 , 5 , 7 , 11 , 13 We know that, For any three sets A , B a n d C , A - B ∪ C = A - B ∩ A - C is called De Morgan's law for set difference. The intersection of two sets A a n d B is the set of all elements which are common to both A a n d B First, we find Q ∩ R = 3 , 5 , 7 Then, P - Q ∩ R = 1 , 2 , 4 , 6 , 8 , 9 . . . . . . 1 The difference of two sets A a n d B is defined as the lists of all the elements that are in set A but that are not present in set B . Next, P - Q = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 - 1 , 3 , 5 , 7 , 9 P - Q = 2 , 4 , 6 , 8 P - R = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 - 2 , 3 , 5 , 7 , 11 , 13 P - R = 1 , 4 , 6 , 8 , 9 The union of sets A a n d B is the set of all elements which are either in A or B or both. Now, P - Q ∪ P - R = 1 , 2 , 4 , 6 , 8 , 9 . . . . . 2 Hence from 1 and 2 , it is verified that P - Q ∩ R = P - Q ∪ P - R .

Verify A ∪ B ' = A ' ∩ B ' using Venn diagrams.

Given to verify: A ∪ B ' = A ' ∩ B ' using Venn diagram We know that, Venn diagram are the diagrams that are used to represent the sets, relation between the sets and operation performed on them, in a pictorial way Let U be the universal set containing finite sets A a n d B , then A ∪ B ' = A ' ∩ B ' is called De Morgan's law for complementation. The given law can verify using Venn diagram as shown below: ......... I ....... I I From I a n d I I A ∪ B ' = A ' ∩ B ' is verified Let us consider an example: U = 0 , 1 , 2 , 3 , 4 , 5 , 6 , A = 1 , 3 , 5 , B = 0 , 3 , 4 , 5 The union of sets A a n d B is the set of all elements which are either in A or B or both. A ∪ B = 1 , 3 , 5 ∪ 0 , 3 , 4 , 5 A ∪ B = 0 , 1 , 3 , 4 , 5 The complement of a set A is the set of all elements of U that are not in A A ∪ B ' = U - A ∪ B = 2 , 6 ..... 1 The intersection of two sets A a n d B is the set of all elements which are common to both A a n d B A ' = U - A = 0 , 2 , 4 , 6 B ' = U - B = 1 , 2 , 6 A ' ∩ B ' = 0 , 2 , 4 , 6 ∩ 1 , 2 , 6 = 2 , 6 ....... 2 From 1 and 2 , it is verified that A ∪ B ' = A ' ∩ B ' .