Algebraic Proofs Flashcards

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Questions and Answers

What is the definition of 'given'?

A process of reasoning

An example

A conclusion

A hypothesis (correct)

What does 'prove' mean?

A statement following IF

A negation

A process of reasoning

A conclusion (correct)

What is the Reflexive Property?

If A, then A=A

Explain the Symmetric Property.

If A=B, then B=A Signup and view all the answers

What does the Distributive Property state?

If A(B+C), then AB+AC Signup and view all the answers

Describe the Substitution Property.

If A=X, then X can be substituted for A Signup and view all the answers

What is the Transitive Property?

If A=B, and B=C, then A=C Signup and view all the answers

Explain the Addition Property.

If A=B, then A+X=B+X Signup and view all the answers

What is the Multiplication Property?

If A=B, then A×B=B×A Signup and view all the answers

Describe the Division Property.

If A=B, then A/X=B/X Signup and view all the answers

What is a conjecture?

A statement that gives an example that follows the hypothesis, but not the conclusion Signup and view all the answers

Define inductive reasoning.

A process that includes looking for patterns and making conjectures, which are unproven statements based on observations Signup and view all the answers

What are conditional statements?

If-Then statements Signup and view all the answers

Explain what a converse is.

Interchanging the hypothesis and the conclusion of a conditional statement Signup and view all the answers

What is an inverse in logic?

Formed by negating both the hypothesis and conclusion Signup and view all the answers

Define contrapositive.

Formed by negating both the hypothesis and conclusion of the converse of a statement Signup and view all the answers

Provide an example of If-Then form.

If it is snowing, then it is cold Signup and view all the answers

What is a hypothesis?

A phrase following, but not including IF Signup and view all the answers

Define conclusion.

The phrase following but not including THEN Signup and view all the answers

What is negation?

The denial of a statement Signup and view all the answers

What are equivalent statements?

Statements that are the same or equal the same thing Signup and view all the answers

Define perpendicular lines.

Two lines that intersect to form right angles Signup and view all the answers

What is a bi-conditional statement?

A statement defined to be true whenever both parts are true Signup and view all the answers

Explain deductive reasoning.

A process that uses facts, definitions, accepted properties, and the laws of logic to form a logical argument Signup and view all the answers

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.

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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.