Question

Let A = {0,2,4,6} B = {1,2,3,4,5} and C = {2,4,6,8}
Verify:- (A ∪ B ) ∪ C = A ∪ (B ∪ C)

Answer 1

Step-by-step explanation:

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

(i) Show that A∪(B∩C)=(A∪B)∩(A∪C).

(ii) Verify this relation using Venn diagram.

Medium

Answer

First, we find A∪(B∩C)

Consider B∩C= {1, 2, 3, 4, 5 ,6} ∩ {2, 4, 6, 7} = {4,6}

A∪(B∩C) = {0,1,2,3,4} U = {4, 6} , = { 0, 1, 2, 3, 4, 6}. (1)

Next, consider A∪B = {0,1, 2, 3, 4} ∪ {1, -2, 3, 4, 5, 6}

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

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

Thus, (A∪B)∩(A∪C) = {-2, 0, 1, 2, 3, 4, 5, 6} ∩ {0, 1, 2, 3, 4, 6, 7}

= {0, 1, 2, 3, 4, 6}. (2)

From (1) and (2), we get A∪(B∩C)=(A∪B)∩(A∪C)

(ii) Using Venn diagram, we have

From (2) and (5) it is verified that A∪(B∩C)=(A∪B)∩(A∪C).

Answer 2

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

(i) Show that A∪(B∩C)=(A∪B)∩(A∪C).

(ii) Verify this relation using Venn diagram.

Medium

Answer

First, we find A∪(B∩C)

Consider B∩C= {1, 2, 3, 4, 5 ,6} ∩ {2, 4, 6, 7} = {4,6}

A∪(B∩C) = {0,1,2,3,4} U = {4, 6} , = { 0, 1, 2, 3, 4, 6}. (1)

Next, consider A∪B = {0,1, 2, 3, 4} ∪ {1, -2, 3, 4, 5, 6}

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

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

Thus, (A∪B)∩(A∪C) = {-2, 0, 1, 2, 3, 4, 5, 6} ∩ {0, 1, 2, 3, 4, 6, 7}

= {0, 1, 2, 3, 4, 6}. (2)

From (1) and (2), we get A∪(B∩C)=(A∪B)∩(A∪C)

(ii) Using Venn diagram, we have

From (2) and (5) it is verified that A∪(B∩C)=(A∪B)∩(A∪C).