1.

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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