1. ## functions

Good morning. Please i need a help to solve the following exercises:

1. Let f: Q --> Q be an homomorfism. Prove that, or f(x)=0 for all x in Q or then f(x) =x for all x in Q

2. Let A, B be sets of positive reals numbers. Letīs define A.B={x.y; x belong to A and y belong to B}. Prove that if A and B are limited, then A.B is limited, where supreme (A.B)=suprmeA.supremeB and inf(A.B)=infA.infB

2. Originally Posted by user
Good morning. Please i need a help to solve the following exercises:

1. Let f: Q --> Q be an homomorfism. Prove that, or f(x)=0 for all x in Q or then f(x) =x for all x in Q
For any homomorphism f, we must have either f(1)= 0 or or f(1)= 1. Look at f(n)= nf(1) and nf(m/n)= f(n(m/n))= mf(1).

2. Let A, B be sets of positive reals numbers. Letīs define A.B={x.y; x belong to A and y belong to B}. Prove that if A and B are limited, then A.B is limited, where supreme (A.B)=suprmeA.supremeB and inf(A.B)=infA.infB
For all x in A and y in B, x< sup A and b< sup B so xy< sup A.sup B. Thus, sup A.sup B is an upper bound on A.B. Suppose there were an upper bound on A.B less than supA.supB. Show that there must be an upper bound on A less than sup A or an upperbound on B less than sup B.

By the way, since I note you are from Costa Rica, some notes on English:
You "need help", not "need a help", the word, in English, is "homomophism", the phrase is "either... or...", not "or ... or...", and sets are "bounded" not "limited".

Still, your English is far better than my Spanish!