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.

    <p>If A=B, then B=A</p> Signup and view all the answers

    What does the Distributive Property state?

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

    Describe the Substitution Property.

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

    What is the Transitive Property?

    <p>If A=B, and B=C, then A=C</p> Signup and view all the answers

    Explain the Addition Property.

    <p>If A=B, then A+X=B+X</p> Signup and view all the answers

    What is the Multiplication Property?

    <p>If A=B, then A×B=B×A</p> Signup and view all the answers

    Describe the Division Property.

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

    What is a conjecture?

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

    Define inductive reasoning.

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

    What are conditional statements?

    <p>If-Then statements</p> Signup and view all the answers

    Explain what a converse is.

    <p>Interchanging the hypothesis and the conclusion of a conditional statement</p> Signup and view all the answers

    What is an inverse in logic?

    <p>Formed by negating both the hypothesis and conclusion</p> Signup and view all the answers

    Define contrapositive.

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

    Provide an example of If-Then form.

    <p>If it is snowing, then it is cold</p> Signup and view all the answers

    What is a hypothesis?

    <p>A phrase following, but not including IF</p> Signup and view all the answers

    Define conclusion.

    <p>The phrase following but not including THEN</p> Signup and view all the answers

    What is negation?

    <p>The denial of a statement</p> Signup and view all the answers

    What are equivalent statements?

    <p>Statements that are the same or equal the same thing</p> Signup and view all the answers

    Define perpendicular lines.

    <p>Two lines that intersect to form right angles</p> Signup and view all the answers

    What is a bi-conditional statement?

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

    Explain deductive reasoning.

    <p>A process that uses facts, definitions, accepted properties, and the laws of logic to form a logical argument</p> 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.

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