If A = {x : x is a natural number}, B = {x : x is an even natural number}, C = {x : x is an odd natural number} and D = {x : x is a prime number}, Find :
(i) `A cap B`
(ii) `A cap C`
(iii) `A cap D`
(iv) `B cap C`
(v) `B cap D`
(vi) `C cap D`.

Explanation

(i) A = {x : x is a natural number}
`={1,2,3,4,5,6,7,8,9,10,11,12,…}`
B = {x : x is an even number} `= {2,4,6,8,10,12,…}`
`:. A cap B = {2,4,6,8,10,12,…}`
= B
Therefore, `A cap B = B`
(ii) A = {x : x is a natural number}
`= {1,2,3,4,5,6,7,8,9,10,12,....}`
C = {x : x is an odd natural number}
`= {1,3,5,7,9,11,...}`
`:. A cap C = {1,3,5,7,9,11,...}`
`= C`
Therefore, `A cap C = C`
(iii) A = {x : x is a natural number}
`= {1,2,3,4,5,6,7,8,9,10,11,12,....}`
D = {x : x is a prime number}
`= {2,3,5,7,11,13,...}`
`:. A cap D = {2,3,5,7,11,....}`
= D
Therefore, `A cap D = D`
(iv) B = {x : x is an even natural number}
`= {2,4,6,8,10,12,...}`
C = {x : x is an odd natural number}
`= {1,3,5,7,9,11,13,..}`
`:. B cap C = phi`
Therefore, `B cap C = phi`
(v) B = {x : x is an even natural number}
`={2,4,6,8,10,12,...}`
D = {x : x is a prime number}
`= {2,3,5,7,11,13,...}`
`:. B cap D ={2}`
Therefore, `B cap D = {2}`
(vi) C = {x : x is an off natural number}
`= {1,3,5,7,...}`
D = {x : x is a prime number}
`= {2,3,5,7,....}`
`:. C cap D` = {x : x is a prime odd natural number}
Therefore, `C cap D = {3,5,7,11,....}`.