Question

Verify De Morgan's laws for the following sets.
(1). U= {x|x€NA15x520)
A= {2, 3, 5, 7, 11, 12, 13, 17}
B = {1, 4, 6, 8, 10, 14, 17, 18}
(ii). If U={1, 2, 3, .... 10}
A = {2, 4, 6, 8, 10
B = {1, 3, 5, 7, 9)
answer​

Answer 1

Step-by-step explanation:

Solution :

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

A = {2, 3}

B = {3, 4, 5}

A ∪ B = {2, 3} ∪ {3, 4, 5}

= {2, 3, 4, 5}

∴ (A ∪ B) ' = {1, 6}

Also A ' = {1, 4, 5, 6}

B ' = {1, 2, 6}

∴ A ' ∩ B ' = {1, 4, 5, 6} ∩ {1, 2, 6}

= {1, 6}

Hence (A ∪ B) ' = A ' ∩ B '

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2) If ξ = {a,b,c,d,e}, A = { a,b,d} and B = {b,d,e}. Prove De Morgan's law of intersection.

Solution :

ξ = {a,b,c,d,e}

A = { a,b,d}

B = {b,d,e}

(A ∩ B) = { a,b,d} ∩ {b,d,e}

(A ∩ B) = {b,d}

∴ (A ∩ B)' = {a, c,e} ----->(1)

A' = {c,e} and B' = {a,c}

∴ A' ∪ B' = {c,e} ∪ {a,c}

A' ∪ B'= { a, c,e} ----->(2)

From (1) and (2)

(A ∩ B)' = A' ∪ B' (which is a De Morgan's law of intersection).