# Introduction to proofs

## Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational

As in previous cases, we start by expressing the conjecture as a conditional statement. If m is irrational and n is rational, then m+n is irrational. In this case, we must use proof by contradiction. Remember that a contradiction is a compound proposition that is always false. To give a proof by contradiction of a …

## Use a direct proof to show that the sum of two even integers is even

To use this method, we must prove that a theorem, stated as a conditional statement p-> q is true. In this case, we can restate the conjecture as follows: If m and n are even integers, then m+n is even. If we assume that p is true (m and n are evens), then we have …

## Use a direct proof to show that the sum of two odd integers is even

Direct proof is one the easiest method to construct proofs. To use this method, we must prove that a theorem, stated as a conditional statement p-> q is true. To this, we assume that p is true, and we apply rules of inference, axioms, definitions and previously proved theorems to prove that q is also …