1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove: (x ≤ y) → (x+z ≤ y+z)

  1. Mar 17, 2012 #1
    [LOGIC] Prove: (x ≤ y) → (x+z ≤ y+z)

    I need to prove if x≤y then x+z ≤ y+z (for all x, y and z)

    Using these axioms (The first 17 are Tarski Arithmetic, and the following 7 are previously proved results)


    All I can think of so far is using Axiom TA16, but then what?

    Last edited: Mar 17, 2012
  2. jcsd
  3. Mar 17, 2012 #2


    User Avatar
    Science Advisor

    Yes, that is one way to do it. Note that what you want to end with, [itex]x+z\le y+ z[/itex] is, again by TA16, equivalent to [itex]0\le (y+ z)- (x+ z)[/itex]. Do you see how to get to that?
  4. Mar 17, 2012 #3

    Got that one now

    For another question on this example sheet I've almost done it apart from the last step where I have to show

    y + (-x) = 0 → y = x

    It seems so simple yet I can't think how to show that, any ideas?
  5. Mar 17, 2012 #4
    Basically I think I need to prove the lemma

    x + z = y + z -> x = y
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook