Question

From the given diagram, find: 
(i) A’ 
(ii) B’ 
(iii) A' ∪ B' 
(iv) (A ∩ B)'

Is A' ∪ B' = (A ∩ B)' ?

Also, verify if A' ∪ B' = (A ∩ B)'.

Answer 1

(i) A = {1, 3, 4, 6}

A' = {2, 5, 7, 8, 9, 10}

(ii) B = {1, 2, 5}

∴ B' = {3, 4, 6, 7, 8, 9, 10}

(iii) A' ∪ B' = {2, 5, 7, 8, 9, 10} ∪ {3, 4, 6, 7, 8, 9, 10}

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

(iv) A ∩ B = {1, 3, 4, 6} ∩ {1, 2, 5}
= {1}

∴ (A ∩ B)' = {2, 3, 4, 5, 6, 7, 8, 9, 10}

From Part (iii) and Part (iv) we conclude

A' ∪ B' = (A ∩ B)'

Now A ∩ B = {2, 5, 7, 8, 9, 10} ∩ {3, 4, 6, 7, 8, 9, 10}

⇒ A' ∪ B' =  {7, 8, 9, 10}       ...(I)

Now A ∪ B = {1, 3, 4, 6} ∪  {1, 2, 5}

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

∴ (A ∩ B)' = {7, 8, 9, 10}      ...(II)

From I and II we conclude

A' ∪ B' = (A ∩ B)'