Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

I'm designing a logic circuit and I was wondering if the following two things are equivalent:

(x <= y) and (x < (y + 1))

I can't think of any set of numbers that would make these two things not equivalent. Only real integers are allowed. Thank you.

share|improve this question
    
Why wont you just test it in Mathematica? –  ALEXANDER Mar 30 at 22:26
    
Implies[(x <= y) , (x < (y + 1))] // Simplify –  rasher Mar 30 at 22:28
    
Have a look at ForALL as well. –  ALEXANDER Mar 30 at 22:31
    
@rasher Could you to justify your claim? I'm affraid that is an artifact of the system even though that is true, because Simplify is not a substitute for Resolve. –  Artes Mar 30 at 23:05
    
@Artes: perhaps I'm mistaken, seem to recall resolve/reduce part of simplify under the covers. –  rasher Mar 31 at 0:10

1 Answer 1

In Mathematica:

    Reduce[ForAll[{x, y}, x <= y \[Equivalent] x < y + 1], Integers]
(* True *)

or:

    Resolve[ForAll[{x, y}, x <= y \[Equivalent] x < y + 1], Integers]
(* True *)

For a mathematical proof, the direct implication $x \leq y \Rightarrow x < y+1$ has the trivial proof noted by @rcollyer in a comment. The converse implication $x < y + 1 \Rightarrow x<=y$ is a consequence of the proposition that, for every integer $y$, there is no integer strictly between $y$ and $y + 1$. The proof of that reduces to the case of nonnegative $y$, and then one may argue from the axioms for the natural numbers.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.