Logic and Proof Concepts Quiz

Vocabulary and Concepts

• Biconditional: A statement that combines a conditional statement and its converse, true if both parts are true.
• Counterexample: An example that disproves a conjecture or statement.
• Informal Proof: A non-structured explanation to demonstrate the validity of a statement or conjecture.
• Postulate: A statement accepted as true without proof, serving as a basis for further reasoning.
• Compound Statement: A statement formed by combining two or more statements using logical operators.
• Deductive Argument: An argument where the conclusion logically follows from premises.
• Inverse: A statement created by negating both the hypothesis and conclusion of a conditional statement.
• Proof: A logical argument demonstrating the truth of a statement, based on definitions, axioms, and previously established theorems.
• Conclusion: The final statement derived from evidence or reasoning; the "then" part of a conditional statement.
• Deductive Reasoning: Reasoning that deduces specific results from general principles or rules.
• Law of Detachment: If a conditional statement is true and its hypothesis is true, then the conclusion is also true.
• Conditional Statement: An "if-then" statement that represents a relationship between two propositions.
• Disjunction: A compound statement formed using "or."
• Law of Syllogism: A logical rule that states if a first conditional statement is true and its conclusion is true, the first hypothesis implies the second conclusion.
• Statement: A declarative sentence that is either true or false.
• Conjecture: An educated guess based on observations or patterns without proof.
• Formal Proof: A structured proof using definitions, axioms, and previously proven statements in a logical format.
• Theorem: A statement that has been proven based on previously established statements and accepted truths.
• Conjunction: A compound statement formed by joining two statements with "and."
• Hypothesis: The "if" part of a conditional statement.
• Matrix Logic: A logic system using matrices to evaluate propositions.
• Truth Table: A table showing all possible truth values for a given logical statement.
• Contrapositive: A statement formed by negating both the hypothesis and conclusion of a conditional and swapping them.
• Negation: The opposite of a given statement, usually indicated by "not."
• Truth Value: The attribute assigned to a statement based on its truth or falsity.
• Converse: A statement formed by reversing the hypothesis and conclusion of a conditional statement.
• Inductive Reasoning: Reasoning that forms generalizations based on specific observations.
• Paragraph Proof: A proof written in paragraph form that explains the reasoning behind the validity of a statement.
• Two-column Proof: A structured proof that contains statements and reasons organized in two columns.

Reasoning and Proof

• Conjectures and Counterexamples: A conjecture is based on patterns; a counterexample is critical for disproving a conjecture, such as proving that point Q is not between points P and R.
• If-Then Statements: In the conditional "If two planes intersect, then their intersection is a line," the hypothesis is "two planes intersect" and the conclusion is "their intersection is a line."
• Truth Value Determination: The truth value of a conditional statement can be evaluated based on conditions, such as water freezing at specific temperatures.

Laws of Reasoning

• Law of Syllogism Application: This law allows deriving valid conclusions based on two true statements, such as connecting the sun's identity as a star to its motion in space.
• Law of Detachment: Valid conclusions may be drawn from established relationships, confirmed by given conditions and previously established statements.

Postulates and Proofs

• Postulates: Fundamental principles, such as "two points determine a line," which are taken as always true.
• Types of Proofs: Various formats exist for proofs, including paragraph and two-column proofs, each emphasizing logical organization and justification of statements.

Angles and Their Relationships

• Angle Relationships: The properties of angles can be utilized in proofs, employing supplementary angles and vertical angle relationships to find angle measures and deduce conclusions.