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Thread: groups qustion

  1. #1
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    groups qustion

    i thought that $\displaystyle \supset$ says that the groups cannot be equal
    but $\displaystyle \supseteq$ says that they can be equal
    so
    why if
    $\displaystyle A \supset B $ then $\displaystyle A \supseteq B
    $
    ??

    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 $\displaystyle A\supseteq B$ and Q be $\displaystyle A\ne B$. Then $\displaystyle A\supset B$ is P and Q. Therefore, $\displaystyle A\supset B$ implies $\displaystyle A\supseteq B$.
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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 $\displaystyle A\supseteq B$ and Q be $\displaystyle A\ne B$. Then $\displaystyle A\supset B$ is P and Q. Therefore, $\displaystyle A\supset B$ implies $\displaystyle A\supseteq B$.
    And to put it another way, $\displaystyle A\supset B$ is a stronger statement than $\displaystyle A\supseteq B$ 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 $\displaystyle A\subset B$ then $\displaystyle A\subseteq B$ is true.

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

    $\displaystyle p \to p\lor q$

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

    Of course $\displaystyle p\lor q$ is just $\displaystyle A \supseteq B$ 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 $\displaystyle A\subset B$ then $\displaystyle A\subseteq B$ is true.

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

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

    Tonio
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