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Math Help - groups qustion

  1. #1
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    groups qustion

    i thought that  \supset says that the groups cannot be equal
    but  \supseteq says that they can be equal
    so
    why if
    A \supset B  then A \supseteq B <br />
    ??

    the one have equality the other doesnt have
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  2. #2
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    For any two propositions (i.e., statements that can be either true or false) P and Q, we have that the compound proposition "P and Q" implies P. This is just a fancy way to describe the meaning of the word "and".

    Now, let P be A\supseteq B and Q be A\ne B. Then A\supset B is P and Q. Therefore, A\supset B implies A\supseteq B.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    For any two propositions (i.e., statements that can be either true or false) P and Q, we have that the compound proposition "P and Q" implies P. This is just a fancy way to describe the meaning of the word "and".

    Now, let P be A\supseteq B and Q be A\ne B. Then A\supset B is P and Q. Therefore, A\supset B implies A\supseteq B.
    And to put it another way, A\supset B is a stronger statement than A\supseteq B in the same way that "XYZW is a square" is stronger than "XYZW is a rectangle".
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  4. #4
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    I would put it another way.

    If A\subset B then A\subseteq B is true.

    If A\subseteq B then A\subset B is false.
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  5. #5
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    Well we probably have enough ways, but here's another one:

    p \to p\lor q

    is a tautology. Let \,p be A \supset B and \,q be A = B.

    Of course p\lor q is just A \supseteq B by definition.
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  6. #6
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    Quote Originally Posted by Plato View Post
    I would put it another way.

    If A\subset B then A\subseteq B is true.

    If A\subseteq B then A\subset B is false.

    I'm guessing you actually meant ( A \subseteq B\Longrightarrow A\subset B ) is false...

    Tonio
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