Reasoning and Proofs Study Notes

Review #2: Reasoning & Proofs

Part 1: Vocabulary

• Theorem: A statement that can be proven and subsequently used as a justification in other proofs.

• Deductive Reasoning: The systematic process of employing logic to derive conclusions from an established set of facts, definitions, and properties.

• Counterexample: A specific instance or case that disproves a conjecture or demonstrates its falsehood.

• Conjecture: A statement or conclusion believed to be true, formulated on the basis of inductive reasoning (observations of patterns).

• Biconditional Statement: A logical statement where both the conditional statement and its converse are true. It is typically phrased as "P if and only if Q" ($P ext{ iff } Q$).

Part 2: Inductive Reasoning

Making Conjectures about Patterns

Inductive reasoning involves observing patterns and predicting the next item or completing a general statement based on these observations.

Examples:

• For visual patterns (Questions 6 & 8): To make a conjecture for a visual pattern, one must observe how the shape, number of elements, orientation, or arrangement changes from one item to the next. The