Deductive Reasoning and Proving
Deductive reasoning is oftentimes applied to constructing valid arguments called proofs. A proof is a collection of statements where one of the statements (called conclusion) absolutely follows from other statements (called premises) of the argument. It is a logical argument showing that the truth of the premises guarantees the truth of the conclusion.
- In a direct proof, one assumes that the premises are true, and then deduces the conclusion from them.
- In an indirect proof, one assumes that the statement to be proven is false, and then deduces a statement that contradicts the hypothesis or a known statement that is true. Once a contradiction is obtained, one concludes that the statement that is assumed false must in fact be true.
To directly prove a conditional statement "if p, then q," assume that the hypothesis p is true, then link p with some statements that are considered true to establish the truth of the conclusion q.
To directly prove a biconditional statement, prove the truth value of the two implications "if p, then q" and "if q, then p."
A proof may be constructed in a two-column form or in a paragraph form. You will see examples of such proofs in the succeeding pages.
A two-column proof is composed of a list of statements and reasons that explain why these statements are true. The statements are listed in the first column while the reasons are listed in the second column.
A proof in paragraph form is similar to a two-column proof, but it is written in sentences. In most cases, it is more advisable to write a two-column proof since the steps and reasons are easier to understand.
