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Math Help - functions

  1. #1
    Junior Member
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    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
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  2. #2
    MHF Contributor

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    Quote Originally Posted by user View Post
    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!
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  3. #3
    Junior Member
    Joined
    Oct 2009
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    answer

    Hi thanks for your answer.
    My english is not high level, but i will take your suggestions.
    I will send another problems about analysis.
    Have a nice day
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