Proof of property of given homomorphism

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Given that $f$ to be group homomorphism from $mathbb R^*tomathbb R^*$ and I want proof of property "If $x>0$ then $f(x)>0$". Please give me hint or something so I can proceed further.







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    up vote
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    down vote

    favorite












    Given that $f$ to be group homomorphism from $mathbb R^*tomathbb R^*$ and I want proof of property "If $x>0$ then $f(x)>0$". Please give me hint or something so I can proceed further.







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Given that $f$ to be group homomorphism from $mathbb R^*tomathbb R^*$ and I want proof of property "If $x>0$ then $f(x)>0$". Please give me hint or something so I can proceed further.







      share|cite|improve this question














      Given that $f$ to be group homomorphism from $mathbb R^*tomathbb R^*$ and I want proof of property "If $x>0$ then $f(x)>0$". Please give me hint or something so I can proceed further.









      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 23 at 5:53









      an4s

      2,0632417




      2,0632417










      asked Aug 23 at 5:03









      Vaibhav Kalia

      204




      204




















          2 Answers
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          If $x>0$, then
          $$x=sqrtxsqrtx$$ so $$f(x)=f(sqrtx.sqrtx)=f(sqrtx)f(sqrtx)=Big(f(sqrtx)Big)^2 >0$$




          Additional exercise: Prove in general that every automorphism of $BbbR^*$ maps positive numbers to positive numbers and negative numbers to negative numbers







          share|cite|improve this answer



























            up vote
            0
            down vote













            Hint: For $a in Bbb R^*$, we have $f(a^2) = f(a)^2$






            share|cite|improve this answer




















            • Ok this is homomorphism property but I didn't get how to use it.
              – Vaibhav Kalia
              Aug 23 at 5:14










            • Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
              – Omnomnomnom
              Aug 23 at 5:26










            • If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
              – Vaibhav Kalia
              Aug 23 at 5:31










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            2 Answers
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            2 Answers
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            active

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            up vote
            2
            down vote













            If $x>0$, then
            $$x=sqrtxsqrtx$$ so $$f(x)=f(sqrtx.sqrtx)=f(sqrtx)f(sqrtx)=Big(f(sqrtx)Big)^2 >0$$




            Additional exercise: Prove in general that every automorphism of $BbbR^*$ maps positive numbers to positive numbers and negative numbers to negative numbers







            share|cite|improve this answer
























              up vote
              2
              down vote













              If $x>0$, then
              $$x=sqrtxsqrtx$$ so $$f(x)=f(sqrtx.sqrtx)=f(sqrtx)f(sqrtx)=Big(f(sqrtx)Big)^2 >0$$




              Additional exercise: Prove in general that every automorphism of $BbbR^*$ maps positive numbers to positive numbers and negative numbers to negative numbers







              share|cite|improve this answer






















                up vote
                2
                down vote










                up vote
                2
                down vote









                If $x>0$, then
                $$x=sqrtxsqrtx$$ so $$f(x)=f(sqrtx.sqrtx)=f(sqrtx)f(sqrtx)=Big(f(sqrtx)Big)^2 >0$$




                Additional exercise: Prove in general that every automorphism of $BbbR^*$ maps positive numbers to positive numbers and negative numbers to negative numbers







                share|cite|improve this answer












                If $x>0$, then
                $$x=sqrtxsqrtx$$ so $$f(x)=f(sqrtx.sqrtx)=f(sqrtx)f(sqrtx)=Big(f(sqrtx)Big)^2 >0$$




                Additional exercise: Prove in general that every automorphism of $BbbR^*$ maps positive numbers to positive numbers and negative numbers to negative numbers








                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 23 at 8:49









                Chinnapparaj R

                1,916318




                1,916318




















                    up vote
                    0
                    down vote













                    Hint: For $a in Bbb R^*$, we have $f(a^2) = f(a)^2$






                    share|cite|improve this answer




















                    • Ok this is homomorphism property but I didn't get how to use it.
                      – Vaibhav Kalia
                      Aug 23 at 5:14










                    • Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                      – Omnomnomnom
                      Aug 23 at 5:26










                    • If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                      – Vaibhav Kalia
                      Aug 23 at 5:31














                    up vote
                    0
                    down vote













                    Hint: For $a in Bbb R^*$, we have $f(a^2) = f(a)^2$






                    share|cite|improve this answer




















                    • Ok this is homomorphism property but I didn't get how to use it.
                      – Vaibhav Kalia
                      Aug 23 at 5:14










                    • Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                      – Omnomnomnom
                      Aug 23 at 5:26










                    • If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                      – Vaibhav Kalia
                      Aug 23 at 5:31












                    up vote
                    0
                    down vote










                    up vote
                    0
                    down vote









                    Hint: For $a in Bbb R^*$, we have $f(a^2) = f(a)^2$






                    share|cite|improve this answer












                    Hint: For $a in Bbb R^*$, we have $f(a^2) = f(a)^2$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Aug 23 at 5:08









                    Omnomnomnom

                    122k784170




                    122k784170











                    • Ok this is homomorphism property but I didn't get how to use it.
                      – Vaibhav Kalia
                      Aug 23 at 5:14










                    • Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                      – Omnomnomnom
                      Aug 23 at 5:26










                    • If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                      – Vaibhav Kalia
                      Aug 23 at 5:31
















                    • Ok this is homomorphism property but I didn't get how to use it.
                      – Vaibhav Kalia
                      Aug 23 at 5:14










                    • Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                      – Omnomnomnom
                      Aug 23 at 5:26










                    • If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                      – Vaibhav Kalia
                      Aug 23 at 5:31















                    Ok this is homomorphism property but I didn't get how to use it.
                    – Vaibhav Kalia
                    Aug 23 at 5:14




                    Ok this is homomorphism property but I didn't get how to use it.
                    – Vaibhav Kalia
                    Aug 23 at 5:14












                    Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                    – Omnomnomnom
                    Aug 23 at 5:26




                    Another hint: if $a in Bbb R^*$, which numbers can $a^2$ be?
                    – Omnomnomnom
                    Aug 23 at 5:26












                    If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                    – Vaibhav Kalia
                    Aug 23 at 5:31




                    If a in R* then $a^2$ will be greater than zero and since f(a^2)=(f(a))^2 this says f(a^2)>0. Correct?
                    – Vaibhav Kalia
                    Aug 23 at 5:31

















                     

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