Using Indirect or Direct Proof in Straightforward Examples

Understanding Indirect and Direct Proof

Direct proof is a method of demonstrating the validity of a statement or theorem by a straight sequence of logical, inferential steps based directly on the axioms, definitions, and previously established statements.
Indirect proof, also known as proof by contradiction or reductio ad absurdum, begins by assuming the opposite of what you want to prove. If this leads to a contradiction, then the original statement must be true.
Both direct and indirect proof are powerful tools in algebra, number theory, and many other areas of mathematics.

Identify the proposition you want to prove. This could be a statement, equation, or inequality.
Apply relevant definitions or axioms. For instance, if the proposition involves even numbers, you might use the definition of an even number.
Use previously established statements or theorems.
Demonstrate the truth of the proposition through a series of logical steps.

Start by assuming the opposite of what you want to prove. If you’re trying to prove a statement is true, start by assuming it’s false.
Use logical reasoning, definitions, axioms, and known theorems to derive a conclusion based on this assumption.
If you reach an absurd result or a contradiction, this means your original assumption must be false. Consequently, the statement you were trying to prove is true.

Use direct proof when you can easily demonstrate a statement’s truth step by step. Direct proof is often simpler and more straightforward.
Use indirect proof when a direct approach is not clear or when the contrary assumption quickly leads to a contradiction. Indirect proofs can often be used to prove statements that are intuitively obvious but difficult to demonstrate directly.

Direct Proof examples: Proving an even plus an even is always an even, proving that the square of an odd number is always odd.
Indirect Proof examples: Proving that there are infinitesimal many prime numbers, proving the irrationality of the square root of 2.

Remember that developing a knack for proofs is a critical part of mathematical maturity and requires consistent practice and engagement.