Question

A = {4, 5, 7, 8, 10}, B = {4, 5, 9} and C = {1, 4, 6, 9},then verify that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

Answer 1

Topic:-

Sets and relations :-

Given :-

• A ={ 4, 5, 7, 8, 10 }
• B = { 4, 5, 9 }
• C = { 1, 4, 6, 9 }

To verify :-

A∩(B ∪ C ) = (A∩B)∪ (A∩C )

Solution :-

Take L.H.S

A∩(B ∪ C )

First we find (B ∪ C )  

B ={ 4, 5, 9}

C ={ 1, 4, 6, 9}

B ∪ C = {4, 5, 9} ∪ { 1, 4, 6, 9}

B ∪C ={ 1, 4, 5, 6, 9}

A∩(B ∪ C )

A = { 4, 5, 7, 8, 10 }

B ∪C ={ 1, 4, 5, 6, 9}

A∩(B ∪ C )  =  { 4, 5, 7, 8, 10 } ∩ { 1, 4, 5, 6, 9}

A∩(B ∪ C ) ={4, 5}

L.H.S = {4,5}

Take R.H.S

(A∩B)∪ (A∩C )

• A ={ 4, 5, 7, 8, 10 }
• B = { 4, 5, 9 }
• C = { 1, 4, 6, 9 }

A∩B = { 4, 5, 7, 8, 10} ∩ { 4, 5, 9}

A ∩ B = { 4, 5}

(A∩C ) = {4, 5, 7, 8,1 0} ∩ { 1, 4, 6, 9}

(A∩C ) = {4}

(A∩B)∪ (A∩C ) = {4, 5} ∪{4}

(A∩B)∪ (A∩C ) = {4, 5}

R.H.S = {4, 5}

L.HS = R. H.S

So,

A∩(B ∪ C ) = (A∩B)∪ (A∩C )

Verified!

Know  more :-

How to find the intersection and union of a given set ?

Intersection means :- If the two sets were given the common elements that present in both are called intersection (∩) symbol

Eg :-

A = {1,2, 3, 4}

B = {0,1,2,3,4}

A∩ B = { 1, 2, 3, 4} i.e these elements are present in both sets i.e called as common elements

Union of a set :- If two sets were given  A, B The elements should belongs to A or B or both A nd B (U) symbol

Eg :-

A = {1,2, 3, 4}

B = {0,1,2,3,4}

A U B = { 0, 1, 2,3 4} i.e both elements of A, B are present

Answer 2

To verify the given equality, we need to show that each set is a subset of the other:

First, we will show that A ∪ (B ∩ C) is a subset of (A ∪ B) ∩ (A ∪ C):

Let x be an arbitrary element of A ∪ (B ∩ C). Then, x is either an element of A or an element of both B and C.

Case 1: x is an element of A In this case, x is an element of (A ∪ B) and (A ∪ C), so it is also an element of (A ∪ B) ∩ (A ∪ C).

Case 2: x is an element of both B and C In this case, x is an element of B ∩ C, and therefore it is also an element of A ∪ (B ∩ C). Since x is an element of both B and C, it is also an element of both A ∪ B and A ∪ C, and thus it is also an element of their intersection (A ∪ B) ∩ (A ∪ C).

Therefore, we have shown that A ∪ (B ∩ C) is a subset of (A ∪ B) ∩ (A ∪ C).

Second, we will show that (A ∪ B) ∩ (A ∪ C) is a subset of A ∪ (B ∩ C):

Let y be an arbitrary element of (A ∪ B) ∩ (A ∪ C). Then, y is an element of both A ∪ B and A ∪ C.

Case 1: y is an element of A In this case, y is an element of A ∪ (B ∩ C).

Case 2: y is an element of both B and C In this case, y is an element of (B ∩ C). Therefore, y is also an element of A ∪ (B ∩ C).

Therefore, we have shown that (A ∪ B) ∩ (A ∪ C) is a subset of A ∪ (B ∩ C).

Since we have shown that each set is a subset of the other, we can conclude that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Hence, the given equality is verified.