The procedure of proof should follow the following format. The answer should like that.

Example

Prove that f :RR?RRsuch that f(x,y) = (2y,-x) is a bijection.

Proof:

(1)Let (a,b) and (c,d) be arbitrary elements ofRRsuch that a?c or b?d.

(2)By the definition of f, f(a,b) = (2b,-a) and f(c,d) = (2d, -c)

(3)Case 1: if a?c, then -a?-c, so (2b,-a)?(2d, -c)

(4)Case 2: if b?d, then 2b?2d, so (2b,-a)?(2d, -c)

(5)By (3) and (4), in all cases we have f(a,b)?f(c,d). This proves that f is one-to-one.

(6)Let (u,v) be an arbitrary element of RR. By definition of f, we have f(-v, u/2) = (u,v)

(7)By (6), (-v,u/2)?f-1(u,v), which shows that f-1(u,v)??. This proves f is onto.

(8)By (5) and (7), f is one-to-one and onto, so f is a bijection

3. For the following question, we only consider subsets of the set R of real numbers. In particular, for any set of real numbers S, we have S-R-s. For each of the following, write out the resulting set using set-builder notation in the style above (i.e., by describing the range(s) of values) (a)?nH (b) GnT