Questions and Answers
What is the definition of 'given'?
What does 'prove' mean?
What is the Reflexive Property?
If A, then A=A
Explain the Symmetric Property.
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What does the Distributive Property state?
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Describe the Substitution Property.
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What is the Transitive Property?
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Explain the Addition Property.
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What is the Multiplication Property?
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Describe the Division Property.
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What is a conjecture?
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Define inductive reasoning.
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What are conditional statements?
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Explain what a converse is.
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What is an inverse in logic?
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Define contrapositive.
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Provide an example of If-Then form.
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What is a hypothesis?
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Define conclusion.
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What is negation?
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What are equivalent statements?
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Define perpendicular lines.
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What is a bi-conditional statement?
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Explain deductive reasoning.
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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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