MATHEMATICAL PROOFS

Direct Proof – Starts with known facts and applies logical steps to prove the statement

Ex: If m is even, then m^2 is even

Proof: By definition, m = 2(some number)

Thus we get M^2 = 4(some number)^2 and is therefore even

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Contrapositive Proof – Proves an equivalent statement by negating both the hypothesis and conclusion.

Ex:  If q^2 is divisible by 3, so is q.

Proof: We will prove the contrapositive;

we will prove if q is not divisible by 3, then q^2 is not divisible by 3.

We know that if q is not divisible by 3, then q^2 ≡ 1 (mod 3).

Thus q^2 is not divisible by 3

 

Contradiction Proof- Assumes the opposite of what is to be proven and shows that this leads to a contradiction.

Ex: There is no largest natural number.

Proof: We will prove the opposite and assume there is a largest natural number N

Since n is a natural number, adding 1 gives another natural number

But N+1 is larger than N, contradicting our assumption that N was the largest natural number.

Our assumption led to a contradiction, so it must be false.

Therefore, there is no largest natural number. ✅

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