Practicing Proof Methods — Which proof method do we use if we assume a statement is false and derive a contradiction from it?

The essence of indirect proof (reductio ad absurdum) is that starting from the negation of the statement, we arrive at a contradiction, so the original statement is true.

Proof Methods

Deduction Predicates

One of the most important parts of mathematics and logic is proof. A proof is the process by which we show that a statement is necessarily true. There are several methods for this.

Direct Proof

In a direct proof, we proceed from known true statements through a sequence of logical steps to the statement to be proved.

Premise: Every even number is divisible by two.
To prove: 8 is divisible by two.
Steps: 8 is an even number → therefore divisible by two.

Proof by Contrapositive

The statement "if p, then q" can also be proved by proving its equivalent form "if not q, then not p".

Example: "If a number is divisible by 4, then it is divisible by 2." Contrapositive: "If a number is not divisible by 2, then it is not divisible by 4."

Indirect Proof (by Contradiction)

In an indirect proof, we assume that the statement is false and then derive a contradiction from it. Therefore, the statement must be true.

Example: Prove that √2 is irrational. Assume the opposite: that √2 is rational. Then it can be written as a/b, where a and b are integers with no common divisor. Deriving a contradiction, √2 cannot be rational.

Summary

Direct proof: We reach the desired statement through logical steps.
Contrapositive: We prove the equivalent form of the statement.
Indirect proof: We derive a contradiction from the negation of the statement.

Practice Exercise

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