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Math Help - determine converse is true or false ..easy question

  1. #1
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    determine converse is true or false ..easy question

    Kindly refer to the attached file. I just to confirm the books ans.

    The book Ans : False
    My Ans : True

    Kindly confirm
    TQ

    Sorry if i post on the wrong section
    Attached Thumbnails Attached Thumbnails determine converse is true or false ..easy question-converse.png  
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  2. #2
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    Re: determine converse is true or false ..easy question

    What if x = 6?
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    Re: determine converse is true or false ..easy question

    Quote Originally Posted by Jester View Post
    What if x = 6?
    Ans will be wrong
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    Re: determine converse is true or false ..easy question

    it make me confuse.If i accept the book answer (false e.g x = 6) but when i subtitute x = 72, the answer will be true.

    correct me if i'm wrong
    Last edited by nikconsult; May 10th 2014 at 08:02 AM.
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    Re: determine converse is true or false ..easy question

    The counter example shows the converse is not true.
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    Re: determine converse is true or false ..easy question

    Quote Originally Posted by nikconsult View Post
    it make me confuse
    Do you know what converse means.

    If $x$ is a multiple of twelve then $x$ is a multiple of three. TRUE.

    CONVERSE: If $x$ is a multiple of three then $x$ is a multiple of twelve. FALSE. Example $x=15$.
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    Re: determine converse is true or false ..easy question

    dear ploto & jester

    If i accept the book answer (false e.g x = 6 or 15) but when i subtitute x = 72, the answer will be true.

    correct me if i'm wrong
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    Re: determine converse is true or false ..easy question

    Quote Originally Posted by nikconsult View Post
    dear ploto & jester
    If i accept the book answer (false e.g x = 6 or 15) but when i subtitute x = 72, the answer will be true.
    correct me if i'm wrong
    The statement "If $x$ is a multiple of twelve then $x$ is a multiple of three."
    is true if and only if it holds for every $x$

    The statement "If $x$ is a multiple of three then $x$ is a multiple of twelve." does not hold for every $x$ so it is false.

    Please stand corrected.
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    Re: determine converse is true or false ..easy question

    Quote Originally Posted by Plato View Post
    The statement "If $x$ is a multiple of twelve then $x$ is a multiple of three."
    is true if and only if it holds for every $x$

    The statement "If $x$ is a multiple of three then $x$ is a multiple of twelve." does not hold for every $x$ so it is false.

    Please stand corrected.
    Dear Mr Plato...
    tq for your respond...it mean FALSE ans is correct??
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  10. #10
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    Re: determine converse is true or false ..easy question

    $P \rightarrow Q$, which is read as "If P then Q", is equivalent to $ (\neg P \lor Q)$, which is read as "either not P or Q". This statement is only false when the antecedent, P, is true while the consequent, Q, is false. Where P, Q are well formed formulas.

    In this particular case you have taken the statement, If x is a multiple of 12, then x is a multiple of 3, and correctly converted it into its consequent, If x is a multiple of 3, then x is a multiple of 12. You now have to either prove the statement or find a counterexample (The question asks you to do this when it says "state whether the converse is true or false").

    For example consider $x=9$. Substituting 9 for x in the converse of the original conditional statement we have: If 9 is a multiple of 3, then 9 is a multiple of 12.

    Recall the definition of a multiple; In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if $b = na$ for some integer n. (This is from wikipedia)

    Evaluate the antecedent first, or "9 is a multiple of 3". Since $9=3^2=3*3$ we conclude that 9 is indeed a multiple of 3.

    Evaluating the consequent second, which is "9 is a multiple of 12". We have it, from the definition of multiple that there exists $n \in \mathbb{N}$ such that $12n=9$. Since $n=\frac{9}{12}=\frac{3}{4}$ n is a rational number, which contradicts our assumption $n \in \mathbb{N}$. Hence we are entitled to conclude the negation of the consequent.

    So we have a situation where the antecedent is true, while the consequent is false. Hence the conditional is false.
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