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Let A = {3, 5, 7}, B = {2, 3, 4, 6} and C = {2, 3, 4, 5, 6, 7, 8}

2
## Answers

**Solution**

(i) (A ∩ B)' = A' ∪ B'

L.H.S. = (A ∩ B)'

A ∩ B = {3}

(A ∩ B)' = {2, 4, 5, 6, 7, 8} ……………….. (1)

R.H.S. = A' ∪ B'

A’ = {5, 7, 8}

B’ = {2, 4, 6}

A’∪B’ = {2, 4, 5, 6, 7, 8} ……………….. (2)

From (1) and (2), we conclude that;

(A ∩ B)' = (A' ∪ B')

(ii) (A ∪ B)' = A' ∩ B'

L.H.S. = (A ∪ B)'

A∪B = {2, 3, 4, 5, 6, 7}

(A ∪ B)' = {8} ……………….. (1)

R.H.S. = A' ∩ B'

A' = {2, 4, 6, 8}

B' = {5, 7, 8}

A' ∩ B' = {8} ……………….. (2)

From (1) and (2), we conclude that;

(A ∪ B)' = A' ∩ B'

**(i) (A ∩ B)' = A' ∪ B'**

**L.H.S. ⇒ (A ∩ B)'**

**⇒A ∩ B = {3} **

**(A ∩ B)' = {2, 4, 5, 6, 7, 8} -----(1) **

**R.H.S. ⇒ A' ∪ B'**

**A’ = {5, 7, 8}**

**∴B’ = {2, 4, 6} **

**A’∪B’ = {2, 4, 5, 6, 7, 8} ---- (2) **

**From (1) and (2), we get,**

**∴(A ∩ B)' = (A' ∪ B') **

**(ii) (A ∪ B)' = A' ∩ B' **

**L.H.S. ⇒ (A ∪ B)'**

**A∪B = {2, 3, 4, 5, 6, 7} **

**⇒(A ∪ B)' = {8} ----- (3) **

**R.H.S. ⇒ A' ∩ B'**

**A' = {2, 4, 6, 8} **

**B' = {5, 7, 8} **

**⇒A' ∩ B' = {8} ----(4) **

**From (3) and (4), we get;**

**∴(A ∪ B)' = A' ∩ B'**