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Thread: equivalent statements

  1. #1
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    Question equivalent statements

    Hi,

    I hope someone can help. Could someone please explain to me the reason behind why the following statement is false? The statement is as follows: “If n is a natural number then it may be expressed as a product of primes” is equivalent to “If n isn’t a natural number then it may not be expressed as a product of primes”.

    I figured that it would be true since it follows the logic "if this, then that" and "if not this, then not that". But evidently, the statement is actually false and I don't understand why.

    I just started learning discrete math, so I would really appreciate help.

    Sincerely,
    Olivia
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  2. #2
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    Re: equivalent statements

    Quote Originally Posted by otownsend View Post
    Hi,

    I hope someone can help. Could someone please explain to me the reason behind why the following statement is false? The statement is as follows: “If n is a natural number then it may be expressed as a product of primes” is equivalent to “If n isn’t a natural number then it may not be expressed as a product of primes”.

    I figured that it would be true since it follows the logic "if this, then that" and "if not this, then not that". But evidently, the statement is actually false and I don't understand why.

    I just started learning discrete math, so I would really appreciate help.
    The negation on If P then Q is P and not Q. Note that there is NO IF in the negation.

    The negation of "If n is a natural number then it may be expressed as a product of primes" is
    "there is a natural number that cannot be expressed as a product of primes".
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  3. #3
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    Re: equivalent statements

    In general, the contrapositive of the statement "if A then B" is "if not B then not A". And the contrapositive of a statement is equivalent to the statement. So a statement equivalent to (not the "inverse" of) "If a number is a natural number then can be expressed as a product of primes" is "if a number is can not be expressed as a product of primes then it is not a natural number".

    I figured that it would be true since it follows the logic "if this, then that" and "if not this, then not that".
    There is no such logic! Suppose it is my custom to carry an umbrella every day, whether is raining or not. Then the statement "if it is raining today then I will carry my umbrella" is true because "I will carry my umbrella" every day including when it is raining. But "if it is not raining then I will not carry my umbrella" is not true.

    (The "contrapositive", "if I do not carry my umbrella then it is not raining" is by default true since the hypothesis "I do not carry my umbrella" is never true.)
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