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  1. #1
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    math proof problem help

    .#1 a) If ex = x for some elements e,x belong to S, we say e is a left identity for x; similarly, if xe = x we say e is a right identity for x. Prove that an element is a left identity for one element of S if and only if it is a left identity for every element of S Let S be a non-empty set with a binary operation which is associative and both left and right transitive

    b) Prove that S has a unique identity element

    c) Deduce that S is a group under the given binary operation




    #2.Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
    Last edited by koukou8617; October 9th 2009 at 10:10 PM. Reason: add more info
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  2. #2
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    Quote Originally Posted by koukou8617 View Post
    #1 a) If ex = x for some elements e,x belong to S, we say e is a left identity for x; similarly, if xe = x we say e is a right identity for x. Prove that an element is a left identity for one element of S if and only if it is a left identity for every element of S

    b) Prove that S has a unique identity element

    c) Deduce that S is a group under the given binary operation




    #2.Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n

    Are you missing some info on this question. Because for any S with a binary operation * defined, we can have multiple left identities
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  3. #3
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    Quote Originally Posted by aman_cc View Post
    Are you missing some info on this question. Because for any S with a binary operation * defined, we can have multiple left identities
    I think the question meant to show that there is only one left and right identity element. If that is correct, simply assume that there exist e_1,e_2 \in S such that they are both identity elements. Then,

    e_1e_2 = e_2e_1 = e_1
    but also
    e_1e_2 = e_2e_1 = e_2

    can you finish?
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by koukou8617 View Post
    .#1 a) If ex = x for some elements e,x belong to S, we say e is a left identity for x; similarly, if xe = x we say e is a right identity for x. Prove that an element is a left identity for one element of S if and only if it is a left identity for every element of S Let S be a non-empty set with a binary operation which is associative and both left and right transitive

    b) Prove that S has a unique identity element

    c) Deduce that S is a group under the given binary operation




    #2.Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
    Transitivity is the key - essentially, this means that for all x,y \in S there exists a s_1, s_2 \in S such that x s_1=y and s_2 x=y (or certainly that is my understanding of it applied here - essentially you are taking the action of the semigroup over the semigroup and saying it is transitive).

    So, if ex=x for some element we have then that x s_1=y \Rightarrow ey = exs_1 = xs_1 = y and so it is a left identity for each element in S.

    The proof is analogous for the right identity, and then the result follows from Defunkt's post.
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  5. #5
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    Quote Originally Posted by Defunkt View Post
    I think the question meant to show that there is only one left and right identity element. If that is correct, simply assume that there exist e_1,e_2 \in S such that they are both identity elements. Then,

    e_1e_2 = e_2e_1 = e_1
    but also
    e_1e_2 = e_2e_1 = e_2

    can you finish?
    @Defunkt - Hi-Thanks for your post. I am in doubt if I am following what you wrote. So, let me plz state that and you tell me if I got it correct or not

    S - is a set with a binary operation defined
    e1 - Right Identity of the set => x.e1 = x, for all x in S
    e2 - Left Identity of the set => e2.x = x, for all x in S

    Are we trying to show e1=e2=e (unique identity element in S)?
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  6. #6
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    Quote Originally Posted by aman_cc View Post
    @Defunkt - Hi-Thanks for your post. I am in doubt if I am following what you wrote. So, let me plz state that and you tell me if I got it correct or not

    S - is a set with a binary operation defined
    e1 - Right Identity of the set => x.e1 = x, for all x in S
    e2 - Left Identity of the set => e2.x = x, for all x in S

    Are we trying to show e1=e2=e (unique identity element in S)?
    Hi.

    What I meant was that both e_1,e_2 are left and right identity elements, ie. \forall x \in S, e_1x = xe_1 = x, e_2x = xe_2 = x
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  7. #7
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Defunkt View Post
    Hi.

    What I meant was that both e_1,e_2 are left and right identity elements, ie. \forall x \in S, e_1x = xe_1 = x, e_2x = xe_2 = x
    hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...
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    Quote Originally Posted by Swlabr View Post
    hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...
    Yep - i would agree with you.
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    Quote Originally Posted by Swlabr View Post
    hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...
    You're right. I misread the first post.
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  10. #10
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    i know how to sole question #1

    can you guys tell e how to do this one

    Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
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  11. #11
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by koukou8617 View Post
    i know how to sole question #1

    can you guys tell e how to do this one

    Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
    What have you done on this problem so far?
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  12. #12
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    Quote Originally Posted by Swlabr View Post
    What have you done on this problem so far?

    i have no idea about this one
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