# What does “assume by contradiction that a<=-2” mean?

I want to prove the following statement.

If a+2>0, then a>-2.

What does "assume by contradiction that a<=-2" mean?

• It means this is going to be a proof by contradiction. That means assume something that's known to be false and show how that isn't right and therefore the opposite must be true. – Jim Nov 26 '19 at 4:03
• There is a stack exchange board for math questions (at any level of expertise), which this is: math.stackexchange.com – Stuart F Jan 7 '20 at 15:55

## 1 Answer

A proof by contradiction depends on the law of the excluded middle. The law of the excluded middle, in this situation, claims that either a > -2 is true OR else a <= -2 is true.

Thus, an "assumption by contradiction" is technically a theorem in propositional logic that is not susceptible of proof or disproof.

• You are right that this is about the Law of the Excluded Middle, but I would revise your second sentence to say: "Either a > -2 is true, or else a <= -2 is true." Given the "if ... then ..." claim in OP, it looks like their goal is to prove that, under some hypotheses, a>-2 is true. One way to show this is to assume the logical negation of that, namely that a is NOT less than -2, i.e. a >= -2. – Brendan W. Sullivan Nov 26 '19 at 15:01
• Mathematicians who reject proof by contradiction don’t tend to get very far. – gnasher729 Jan 3 '20 at 12:07
• @gnasher729 But if they did, ... ;) – Phil Sweet Jan 7 '20 at 11:43