Algebraic Proofs Flashcards

Study Notes

Algebraic Proofs Key Concepts

Given: Represents the hypothesis in a mathematical statement, establishing conditions or assumptions to be considered.
Prove: Refers to the conclusion drawn from logical reasoning and the given premises in a proof.

Fundamental Properties

Reflexive Property: States that any quantity is equal to itself (e.g., A = A).
Symmetric Property: Indicates if A equals B, then B also equals A; equality is two-way.
Transitive Property: Connects relationships; if A equals B and B equals C, then A equals C.
Distributive Property: Describes how multiplication distributes over addition (e.g., A(B + C) = AB + AC).
Substitution Property: Allows substitution of quantities for one another based on equality (if A = X, then replace A with X).
Addition Property: States if A equals B, then adding the same term X to both equations preserves equality (A + X = B + X).
Multiplication Property: If A equals B, then multiplying both sides by the same term equates them (A × X = B × X).
Division Property: States that if A equals B, then dividing both sides by the same non-zero term keeps the equality (A / X = B / X).

Reasoning Techniques

Conjecture: A suggestion or statement believed to be true, inferred from examples, but not universally proven.
Inductive Reasoning: Involves identifying patterns and making conjectures based on observed examples; often leads to unproven but plausible statements.
Deductive Reasoning: Utilizes established facts, definitions, and logical principles to formulate valid arguments and proofs.

Statement Forms

Conditional Statement: Formulated as "If-Then"; comprises an hypothesis and a conclusion.
Comverse: The result of reversing the hypothesis and conclusion of a conditional statement.
Inverse: Developed by negating both the hypothesis and conclusion of a conditional statement.
Contrapositive: Created by negating both the hypothesis and conclusion after switching them, forming a new statement based on the converse.

More Definitions

Negation: The opposite or denial of a given statement.
Equivalent Statements: Statements that hold the same truth value, meaning they are interchangeable.
Perpendicular Lines: Two lines that intersect to form right angles, confirming equivalence in their geometric properties.
Bi-conditional Statement: Defined to be true if both constituent statements are true; functions as an "if and only if" condition.
Hypothesis: The segment of a conditional statement following "if" but not including it.
Conclusion: Follows "then" in a conditional statement, representing the result or outcome of the hypothesis.

Description

Test your understanding of important algebraic properties with these flashcards. Each card presents a key term and its definition, helping you to reinforce your knowledge of algebraic proofs. Perfect for students looking to excel in their algebra studies.