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Diagram

Sum

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)'.

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#### Solution

**(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)'

Concept: Venn Diagram

Is there an error in this question or solution?

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