Let x and y be rational numbers. Show that xy is a rational number.

If a and b are positive integers, then you know that a = bq + r, 0 ≤ r < b, where q is a whole number. Prove that HCF (a, b) = HCF (b, r).

In each of the following questions, there are two statements. State if the second is the negation of the first or not.

(i) Mumtaz is hungry ; Mumtaz is not hungry.

(ii) Some cats are black ; Some cats are brown.

(iii) All elephants are huge ; One elephant is not huge.

(iv) All fire engines are red ; All fire engines are not red.

(v) No man is a cow ; Some men are cows.