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If U = {1,2,3,4,5,6,7,8,9,10,12,24} A = {x ∶ x is prime number and x ≤ 10} B = {x ∶ x is a factor of 24} Verify the following result (i) A – B = A ∩ B’ (ii) (A ∪ B)’ = A’ ∩ B’ (iii) (A ∩ B)’ = A’ ∩ B’ |
Answer» Given, U = {1,2,3,4,5,6,7,8,9,10,12,24} A = {2,3,5,7} B = {1,2,3,4,5,6,8,12,24} Now, A’ = {1,4,6,8,9,10,12,24} B’ = {5,7,9,10} A ∪ B = {1,2,3,4,5,6,7,8,12,24} (A ∪ B)’ = {9,10} A ∩ B = {2,3} (A ∪ B)’ = {1,4,5,6,7,8,9,10,12,24} (i) A – B = A ∩ B’ L.H.S = A – B = {2,3,5,7} – {1,2,3,4,6,8,12,24} = {5,7} R.H.S = A ∩ B’ = {2,3,5,7} ∩ {5,7,9,10} = {5,7} ∴ L.H.S = R.H.S, (ii) (A ∪ B)’ = A ∩ B’ L.H.S = (A ∪ B)’ = {9,10} R.H.S = A’∩ B’ = {1,4,6,8,9,10,12,24} ∩ {5,7,9,10} = {9,10} ∴ L.H.S = R.H.S, (iii) (A ∩ B)’ = A’ ∩ B’ L.H.S = (A ∩ B)’ = {1,4,5,6,7,8,9,10,12,24} R.H.S = A’ ∩ B’ = {1,4,6,8,9,10,12,24} ∩ {5,7,9,10} = {1,4,5,6,7,8,9,10,12,34} ∴ L.H.S = R.H.S, |
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