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Math Help - Help with implications

  1. #1
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    Help with implications

    I want to prove that

    a < b \Rightarrow a + c = b + c

    Which of course is not true.

    I don't know how to do this kind of proofs.
    I mean, when trying to prove
    A \Rightarrow B
    should I try to work on the left side of the implication until A is equal to B? Am I allowed to take on what I have on the left side and make a substitution of that on the right side?

    i.e., is this allowed?:
    a < b \Leftrightarrow a + x = b, x > 0

    a < b \Rightarrow a + c = b + c
    a + x = b \Rightarrow a + c = b + c
    a + x = b \Rightarrow a + c = (a + x) + c
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  2. #2
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    Quote Originally Posted by devouredelysium View Post
    I want to prove that

    a < b \Rightarrow a + c = b + c

    Which of course is not true.
    Then what do you mean "I want to prove" it? You can't prove something that isn't true!

    I don't know how to do this kind of proofs.
    I mean, when trying to prove
    A \Rightarrow B
    should I try to work on the left side of the implication until A is equal to B?
    I don't know what you mean by "A is equal to B". In general, the hypothesis and conclusion of a proof (A and B) are statements or combinations of statements and it doesn't make sense to say "equal".

    Am I allowed to take on what I have on the left side and make a substitution of that on the right side?

    i.e., is this allowed?:
    a < b \Leftrightarrow a + x = b, x > 0

    a < b \Rightarrow a + c = b + c
    a + x = b \Rightarrow a + c = b + c
    a + x = b \Rightarrow a + c = (a + x) + c
    I have no idea what you are trying to do here.

    If you want to prove the "a< b" implis "a+ x= b for x> 0" then somewhere you are going to have to use the definition of "a< b". How is that defined in your course?

    One thing that it is important to realize- definitions in mathematics are working definitions- you use the precise words of definitions in proofs.
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  3. #3
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    Quote Originally Posted by devouredelysium View Post
    I want to prove that

    a < b \Rightarrow a + c = b + c

    Which of course is not true.

    I don't know how to do this kind of proofs.
    I mean, when trying to prove
    A \Rightarrow B
    should I try to work on the left side of the implication until A is equal to B? Am I allowed to take on what I have on the left side and make a substitution of that on the right side?
    I don't understand why you want to prove something that you know is false, but I know mathematically to prove P \Rightarrow Q you go from P to Q, and to prove P \Leftrightarrow Q, you must prove P \Rightarrow Q and Q \Rightarrow P

    To prove a < b \Rightarrow a+c = b+ c is false.

    From truth table of implication, we know that if P is true and Q is false, the implication P \Rightarrow Q is false. Based on this logical principle, we proceed as follows:

    Let a,b \in R. By Trichotomy Law, if a,b \in R, then there are three possiblilities, namely a<b, a=b, or a>b.

    Suppose that a<b. Then a\not < b and a \not =b.

    Case 1: Say a\not =b. Adding c to bothside of equality, by principle of arithmetic operation a+c \not= b+c, which in logical term a<b \Rightarrow a+c\not=b+c as desired.

    Case 2: a\not >b. The process is similar.

    It's not an exciting proof.
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