Prove: (x ≤ y) → (x+z ≤ y+z)

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In summary, the conversation is about proving the statement (x ≤ y) → (x+z ≤ y+z) using various axioms and previously proven results. The last step involves showing that y + (-x) = 0 → y = x, which can be done by proving the lemma x + z = y + z → x = y.
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[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)

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

Thanks
 
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  • #2
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?
 
  • #3
Thanks!

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?
 
  • #4
Basically I think I need to prove the lemma

x + z = y + z -> x = y
 

What is the statement "Prove: (x ≤ y) → (x+z ≤ y+z)" asking to prove?

The statement is asking to prove that if x is less than or equal to y, then x+z is also less than or equal to y+z.

What does the symbol "↔" mean in this statement?

The symbol "↔" represents the logical biconditional, which means that the statement is true both ways. In this case, it means that if x ≤ y, then x+z ≤ y+z and if x+z ≤ y+z, then x ≤ y.

What is the difference between "→" and "↔" in this statement?

The symbol "→" represents the conditional, which means that the statement is true if the first part (x ≤ y) is true, regardless of the truth value of the second part (x+z ≤ y+z). The symbol "↔" represents the biconditional, which means that the statement is true both ways - if the first part is true, then the second part is also true, and vice versa.

What is the significance of the condition "x ≤ y" in this statement?

The condition "x ≤ y" is significant because it sets a limit or boundary for the values of x and y. It means that x cannot be greater than y, which affects the truth value of the statement. If x is not less than or equal to y, then the statement cannot be proven true.

How can one prove this statement to be true?

This statement can be proven to be true by using logical reasoning and mathematical principles. One approach would be to use proof by contradiction, where you assume the statement is false (x ≤ y) → (x+z ≤ y+z) is false and show that it leads to a contradiction. Another approach would be to use algebraic manipulation and substitution to show that the statement holds for all values of x and y.

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