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How do I prove an either/or inequality?

  1. Feb 12, 2017 #1
    1. The problem statement, all variables and given/known data
    "Given: ##a,b,c∈ℤ##,
    Prove: If ##2a+3b≥12m+1##, then ##a≥3m+1## or ##b≥2m+1##."

    2. Relevant equations
    ##P:a≥3m+1##
    ##Q:b≥2m+1##
    ##R:2a+3b≥12m+1##

    3. The attempt at a solution
    Goal: ##~(P∨Q)≅(~P)∧(~Q)⇒~R##

    Assume that ##a<3m+1## and ##b<2m+1##. Then ##2a+3b<2(3m+1)+3(2m+1)=12m+5##. But this doesn't necessarily imply that ##2a+3b>12m+1##. Can someone help me connect the dots?
     
  2. jcsd
  3. Feb 12, 2017 #2

    FactChecker

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    The problem doesn't ask you to prove that. Your proof by contradiction is complete as it is.
    CORRECTION: I should have said proof by contrapositive.
    CORRECTION 2: I completely missed the point that the OP proves <12m+5 but it needs <12m+1
     
    Last edited: Feb 13, 2017
  4. Feb 12, 2017 #3
    Hi,

    Your statement must be wrong since ##a=5,b=1## and ##m=1## clearly violate it.
     
  5. Feb 12, 2017 #4

    FactChecker

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    Note. Tildas are difficult to do in LaTeX. I don't know how to do them. Here is a readable version of your Goal.
     
  6. Feb 12, 2017 #5
    I was trying to do proof by contra-positive, but I ended up with an inequality in my original post that says nothing about my original statement. I figured that it was only conditionally true. So I ended up with a statement just saying that the statement is false when at least one of the inequalities is false; then I gave a counter-example.
     
  7. Feb 13, 2017 #6

    Ray Vickson

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    Since ##a, b, m## are all integers, you can re-write the three inequalities in the original question by first removing the "+1" on all three right-hand-sides and replacing "≥" by ">". That works because both sides are integers. It is worth doing---try it and see.
     
  8. Feb 13, 2017 #7

    FactChecker

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    If a < 3m+1 and you are restricted to integers, then you know that a ≤ A = 3m. Likewise b < 2m+1 implies b ≤ B = 2m. Compare 2a+3b with A+B and 12m+1.
    CORRECTION: Should have said "Compare 2a+3b with 2A+3B and 12m+1."
     
    Last edited: Feb 14, 2017
  9. Feb 13, 2017 #8
    Well, that simplified my argument by a whole lot more. Thanks, too, @FactChecker.
     
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