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Thread: proof

  1. #1
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    proof

    If A is a subset of B, then A X A is a subset of B X B

    hoe do you prove this
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  2. #2
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    Quote Originally Posted by leinadwerdna View Post
    If A is a subset of B, then A X A is a subset of B X B

    hoe do you prove this
    $\displaystyle A\subseteq B$ so if $\displaystyle x\in A$ then $\displaystyle x\in B$


    A X A=$\displaystyle \{ (x,y) | x,y\in A \}$ but since $\displaystyle x,y\in A$, $\displaystyle x$ and $\displaystyle y \in B$

    So all $\displaystyle (x,y)\in B$, so AXA$\displaystyle \subseteq $BXB
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  3. #3
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    how do you prove the reverse of that

    if A X A subset B X B, then A subset of B
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  4. #4
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    Quote Originally Posted by leinadwerdna View Post
    how do you prove the reverse of that
    if A X A subset B X B, then A subset of B
    If $\displaystyle x\in A$ then $\displaystyle (x,x)\in A\times A$. Can you explain that?

    Then can you prove that implies that $\displaystyle x\in B?$

    How does that prove that $\displaystyle A \subseteq B?$
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  5. #5
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    Proof: Assume x is an element of A. (x,x) is therefore an element of A X A because the definition of A X A is the set of (x,x) such that x is an element of A and x is an element of A. Since , A X A is a subset of B X B (x,x) must also be an element of B X B which means X is an element of B and x is an element of B. x is an element of B so A is a subset of B

    is this right reasoning
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  6. #6
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    Quote Originally Posted by leinadwerdna View Post
    Proof: Assume x is an element of A. (x,x) is therefore an element of A X A because the definition of A X A is the set of (x,x) such that x is an element of A and x is an element of A. Since , A X A is a subset of B X B (x,x) must also be an element of B X B which means X is an element of B and x is an element of B. x is an element of B so A is a subset of B. Is this right reasoning
    Yes. Good job.
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  7. #7
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    thanks
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