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Math Help - Proving subsets =S

  1. #1
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    Proving subsets =S

    Hello
    Iv got an exam in a week and am trying to get through my whole course in that week. And am off to a horrible start =/
    Can someone please tell how i should go about proving questions like the follwoing.

    Q1)

    A = {x <element_of> R | cos x = 1}
    A = {x <element_of> R | sin x = 0}
    Show that A <subsest of> B


    Q2)

    X = {24k + 7 | k <element_of> Z}
    Y = {4n + 3 | n <element_of> Z}
    Z= {6m + 1 | m <element_of> Z}

    prove: X <subset_of> Y and X <subset_of> Z but Y <not_subset_of> Z

    Thanks and im sure alot more questions would be comming soon =(
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lost1 View Post
    Hello
    Iv got an exam in a week and am trying to get through my whole course in that week. And am off to a horrible start =/
    Can someone please tell how i should go about proving questions like the follwoing.

    Q1)

    A = {x <element_of> R | cos x = 1}
    A = {x <element_of> R | sin x = 0}
    Show that A <subsest of> B


    Q2)

    X = {24k + 7 | k <element_of> Z}
    Y = {4n + 3 | n <element_of> Z}
    Z= {6m + 1 | m <element_of> Z}

    prove: X <subset_of> Y and X <subset_of> Z but Y <not_subset_of> Z

    Thanks and im sure alot more questions would be comming soon =(
    Appeal to the definition of subset.  X \subset Y \Leftrightarrow \left(x\in X\Rightarrow x\in Y\right)

    So for the first one. We want to show that if x\in X\implies\cos(x)=1 that x\in Y\implies\sin\left(x\right)=0.

    Can you do that?
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  3. #3
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    Hey thanks for the quick reply =)

    hmm...yess i tried using the basic idea but couldnt get a proper proof. Would this be a clear enough proof:

    Let x <element_of> A, that is cos x = 1
    ie. x = 2kPi where k <element_of> N

    Now, sin(x) = sin(2kPi) = 0 and therefor x <element_of> B
    That is, A <subset_of> B

    ?
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  4. #4
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    also, how do i show the maths symbols nicely like u have?
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lost1 View Post
    Hey thanks for the quick reply =)

    hmm...yess i tried using the basic idea but couldnt get a proper proof. Would this be a clear enough proof:

    Let x <element_of> A, that is cos x = 1
    ie. x = 2kPi where k <element_of> N

    Now, sin(x) = sin(2kPi) = 0 and therefor x <element_of> B
    That is, A <subset_of> B

    ?
    Did you take a non-decript element of A. Yes. Did you show that this element must be an element of B. Yes. Did you give a complete and correct proof. Yes.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lost1 View Post
    also, how do i show the maths symbols nicely like u have?
    Enclose them in math brackets For example. to do the \underbrace{X\cup Y=\int_0^{\Phi}f_n(x)dx}_{\text{this}} you must merely enter
    Code:
    \underbrace{X\cup Y=\int_0^{\Phi}f_n(x)dx}_{\text{this}}
    this enclosed in [ math] [/tex].
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  7. #7
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    thanks =)
    now how do i go about doing the second one?
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lost1 View Post
    thanks =)
    now how do i go about doing the second one?
    Well 24k+7=4(6k)+4+3=4(6k+1)+3 And clearly 6k+1\in\mathbb{Z}.......so
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  9. #9
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    ohh god i am thick arnt I...
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lost1 View Post
    ohh god i am thick arnt I...
    Nah. Little stupid things like that are always the hardest.
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  11. #11
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    ok another quick question...

    Prove if
    <br />
A <intersect> C \subset B <instersect> C
    and
    <br />
A \cup C \subset B \cup C<br />

    then  A \subset B

    My start at proof:
    Let  x \in A therefore x \in B \cup C

    and then =/

    also whats the code for intersection?
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