Question

Show that the following four conditions are equivalent :
(i) `A sub B`
(ii) `A - B = phi`
(iii) `A cup B = B`
(iv) `A cap B = A`.

Answer 1

(i) Let `A sub B`
`rArr` All elements of A are in B
(ii) `A - B = phi rArr` All elements of set A which are not in `B in phi`
`rArr` There is no element in set A which is not in B
`rArr` All elements of set A are in set B
`rArr A sub B`
`:. A- B = phi rArr A sub B`
(iii) Let `A cup B = B`
`because A sub A cup B`
Therefore, `A sub B` [` because` Given `A cap B = B`]
`:. A cup B = B rArr A sub B`
(iv) Let `A cap B =A`
`because A cap B sub B`
`rArr A sub B` [ `because` Given `A cap B = A`]
`:. A cap B = A rArr A sub B`
Therefore, four conditions are equivalent.