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Math Help - Proofs/ counterexamples

  1. #1
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    Proofs/ counterexamples

    Find a counterexample to show that the converse of each conditional is false.

    1. If x=-5, then x^2=25.

    2. If two angles are adjacent, then they share a common vertex.




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  2. #2
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    Re: Proofs/ counterexamples

    Quote Originally Posted by Christyh101 View Post
    Find a counterexample to show that the converse of each conditional is false.

    1. If x=-5, then x^2=25.
    The converse is \text{If }x^2=25\text{ then }x=-5~.
    You surely are able to supply a counter-example.
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  3. #3
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    Re: Proofs/ counterexamples

    Hello, Christyh101!

    Do you know what adjacent angles are?
    Then you can surely come up with a counterexample . . .


    Find a counterexample to show that the converse of the conditional is false.

    2. If two angles are adjacent, then they share a common vertex.

    The converse is:
    . . If two angles share a common vertex, then they are adjacent.

    Adjacent angles are angles that share a common side.


    The converse is false.
    We can have two angles with a common vertex,
    . . which do not share a common side.


    Code:
          |..../
          |.../
          |A./
          |./
          |/
          *---------
           \.B.....
            \.....
             \...
              \.
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  4. #4
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    Re: Proofs/ counterexamples

    in general (and this is worth thinking about now) when you have a statement like "if A, then B" it means that A is a stronger statement than B (and is true for fewer things).

    for example, x = -5 tells us everything we need to know about x. only one x will make this statement true, namely, -5.

    by contrast, x^2 = 25 only tells us "something" about x, that its square is 25. this happens to be true for two values of x, one is -5, another is 5.

    similarly, A and B are adjacent angles, tells us two pieces of information: they share a vertex, and a common side.

    this is clearly stronger than: A and B share a common vertex, which is only one piece of information (weaker).

    one thing you should have clear in your mind, always: if A is stronger than B, we may not be able to derive A from B.

    IF we can, then A and B are "the same strength", that is, they are equivalent statements (true for the same things).

    so:

    A implies B

    A is stronger, true for fewer (or the same number of) things, more restrictive
    B is weaker, true for more (or the same number of) things, more inclusive

    you will come upon this "shape of thought" in many areas: logical thinking, mathematics (of many kinds), action and consequence. learn it well.
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