Question

If U={1,2,3,4,5}. A={1,2,5} B={3,4,5}. prove that : A'n B' = (A U B)'​

Answer 1

$\large\underline{\sf{Given- }}$

U={1,2,3,4,5}

A={1,2,5}

B={3,4,5}

$\large\underline{\sf{To\:prove - }}$

$\rm :\longmapsto\: A'\cap B' = (A\cup B)'$

$\purple{\large\underline{\sf{Solution-}}}$

Given that,

U={1,2,3,4,5}

A={1,2,5}

B={3,4,5}

Now,

$\rm :\longmapsto\:A' = \{3, \: 4 \}$

$\rm :\longmapsto\:B' = \{1, \: 2\}$

Thus,

$\red{\rm :\longmapsto\: \red{ \boxed{ \sf{ \:A'\cap B' = \phi}}}}$

Now,

$\rm :\longmapsto\:A\cup B = \{1,2,3,4,5\}$

Thus,

$\red{\rm :\longmapsto\: \boxed{ \sf{ \:(A\cup B)' \: = \: \phi}}}$

Thus, we concluded that

$\red{\rm :\longmapsto\: \boxed{ \sf{ \:(A\cup B)' \: = \: A' \: \cap \: B'}}}$

Hence, Proved

Additional Information :-

1. Commutative Law

$\red{ \boxed{ \sf{ \:A\cup B = B\cup A}}}$

$\red{ \boxed{ \sf{ \:A\cap B =B \cap A}}}$

2. Associative Law

$\red{ \boxed{ \sf{ \:(A\cup B)\cup C = A\cup (B\cup C)}}}$

$\red{ \boxed{ \sf{ \:(A\cap B)\cap C = A\cap (B\cap C)}}}$

3. Distributive Law

$\red{ \boxed{ \sf{ \:A\cup (B\cap C) = (A\cup B)\cap (A\cup C)}}}$

$\red{ \boxed{ \sf{ \:A\cap (B\cup C) = (A\cap B)\cup (A\cap C)}}}$

Answer 2

Answer:

U = { 1,2,3,4,5,6}

A = {2,3}

B= {3,4,5}

AUB= { 2,3} U {3,4,5}= { 2,3,4,5}

therefore, (AUB) = { 1,6}

B= {1,2,6}

therefore,AUB= {1,4,5,6} U = {1,2,6}= {1,6}