Question

5) If A= {1,2,3,4}, B = {3,4,5,6},
C = {4, 5, 6, 7, 8} and universal set
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10), then verify
the following:
1)An(BuC)=(AnB)u(AuC)
​

Answer 1

$\large\bf\underline {To \: find:-}$

• we need to verify that :-
• A∩(B ∪ C) = (A ∩ B) ∪ (A ∪ C)

$\large\bf\underline{Given:-}$

• A= {1,2,3,4}
• B = {3,4,5,6}
• C = {4, 5, 6, 7, 8}
• universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

$\huge\bf\underline{Solution:-}$

A= {1,2,3,4}

B = {3,4,5,6}

C = {4, 5, 6, 7, 8}

∪ = X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(B ∪ C) = { 3 ,4 ,5 ,6 ,7 ,8 } .....1)

Intersection from 1) and set A we get,

A∩(B ∪ C) = { 3 ,4 } .....2)

(A ∩ B) = { 3 , 4 } ......3)

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

Now,

Union set from 3) and 4)

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

Now,

✝️ verification :-

A∩(B ∪ C) = (A ∩ B) ∪ (A ∪ C)

{ 3 ,4 } ≠ { 1 ,2 ,3 ,4 ,5 ,6 ,7 ,8}

So,

A∩(B ∪ C) ≠ (A ∩ B) ∪ (A ∪ C)

━━━━━━━━━━━━━━━━━━━━━━━

Answer 2

A= {1,2,3,4}

B = {3,4,5,6}

C = {4, 5, 6, 7, 8}

X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10) - universal set

Finding AuB, BuC and AuC

AnB = {1,2,3,4} n {3,4,5,6}

AnB = {3,4}

BuC = {3,4,5,6} u {4,5,6,7,8}

BuC = {3,4,5,6,7,8}

AuC = {1,2,3,4} u {4,5,6,7,8}

AuC = {1, 2,3,4,5,6,7,8}

LHS

An(BuC) = {1,2,3,4} n {3,4,5,6,7,8}

An(BuC) = {3,4}

RHS

(AnB)u(AuC) = {3,4} u {1,2,3,4,5,6,7,8}

(AnB)u(AuC) = { 1,2,3,4,5,6,7,8

LHS is not equal RHS

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