Geometry and Logic Fundamentals

Comprehensive vocabulary flashcards covering mathematical logic, truth values, compound statements, laws of inference, conditional variants, and foundational geometric concepts.

Open Sentence

A sentence that contains a variable (for example, $x + 7y - 12 = 9$).

Statement (Closed Sentence)

A declarative sentence that is true or false but not both.

Negation

A statement that makes a claim opposite that of the original statement; the negation of statement $p$ is written symbolically as $\sim p$ and is read 'not $p$'.

Logically Equivalent

The relationship between two statements if each statement yields the same truth value in every possible situation.

Law of Double Negation

A logical law stating that the negation of the negation of a statement is logically equivalent to the original statement ($\sim(\sim P) \equiv P$).

Compound Statement

A statement formed by combining two or more statements, such as conjunctions and disjunctions.

Conjunction

A compound statement formed by inserting the word 'and' between two statements, written symbolically as $p \wedge q$ and read as '$p$ and $q$', which is true only when both $p$ and $q$ are true.

Law of Simplification

A law of inference stating that given a true conjunction $P \wedge Q$, one can conclude that $P$ is true (or that $Q$ is true).

Disjunction

A compound statement formed by inserting the word 'or' between two statements, written symbolically as $p \vee q$ and read as '$p$ or $q$', which is true when $p$ is true, $q$ is true, or both are true.

Law of Disjunctive Inference

A law of inference stating that given a disjunction $(P \vee Q)$ and the negation of one component ($\sim P$), one can conclude that the other component ($Q$) is true.

De Morgan’s Laws

Logical rules stating that the negation of a conjunction is equivalent to the disjunction of the negations ($\sim(p \wedge q) \equiv \sim p \vee \sim q$), and the negation of a disjunction is equivalent to the conjunction of the negations ($\sim(p \vee q) \equiv \sim p \wedge \sim q$).

Reasoning

The process based on experience and principles that leads to a conclusion.

Intuition

A type of reasoning consisting of a sudden insight that allows one to make a statement without any formal reasoning.

Induction

A type of reasoning where specific observations and experiments are used to draw a general conclusion.

Deduction

A type of reasoning in which the knowledge and acceptance of selected assumptions guarantee the truth of a particular conclusion.

Conditional Statement

A compound statement of the form 'If $p$, then $q$', written symbolically as $p \rightarrow q$ and read as 'if $p$, then $q$' or '$p$ implies $q$'.

Hypothesis

The statement immediately following the word 'if' in a conditional statement (represented by $p$ in the statement $p \rightarrow q$).

Conclusion

The statement immediately following the word 'then' in a conditional statement (represented by $q$ in the statement $p \rightarrow q$).

Law of Detachment (Modus Ponens)

A law of inference stating that whenever a conditional statement ($p \rightarrow q$) and its hypothesis ($p$) are true, the conclusion ($q$) is also true.

Law of Modus Tollens

A law of inference stating that whenever a conditional statement is true ($p \rightarrow q$) and its conclusion is false ($\sim q$), the hypothesis is false ($\sim p$).

Inverse

The conditional statement formed by negating both the hypothesis and conclusion of a given conditional statement; the inverse of $p \rightarrow q$ is $\sim p \rightarrow \sim q$.

Converse

The conditional statement formed by switching the order of the hypothesis and conclusion of a given conditional statement; the converse of $p \rightarrow q$ is $q \rightarrow p$.

Contrapositive

The conditional statement formed by negating and switching the order of the hypothesis and conclusion of a given conditional statement; the contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$.

Law of Contrapositive

A logical law stating that a conditional statement and its contrapositive are logically equivalent ($p \rightarrow q \equiv \sim q \rightarrow \sim p$).

Biconditional Statement

A compound statement formed by the conjunction of a conditional statement and its converse, written symbolically as $p \leftrightarrow q$ and read '$p$ if and only if $q$'; it is true when $p$ and $q$ have the same truth value and false otherwise.

Tautology

Any symbolic statement that is always true.

Law of Syllogism

A law of inference stating that given two conditional statements $p \rightarrow q$ and $q \rightarrow r$, one can conclude $p \rightarrow r$.

Contradiction

A statement or compound statement that is always false.

Negation of a Conditional Statement

The symbolic statement $\sim(p \rightarrow q)$, which is logically equivalent to $p \wedge \sim q$.

Negation of a Biconditional Statement

The symbolic statement $\sim(p \leftrightarrow q)$, which can be represented as $(p \wedge \sim q) \vee (\sim p \wedge q)$, $\sim p \leftrightarrow q$, or $p \leftrightarrow \sim q$.

Segment Addition Postulate (SAP)

A geometric postulate stating that if $X$ is a point on $\overline{AB}$ and $A\text{-}X\text{-}B$ ($X$ is between $A$ and $B$), then $AX + XB = AB$.

Perpendicular Bisector Construction

A compass-and-straightedge construction that creates a line perpendicular to segment $\overline{AB}$ passing through its midpoint by drawing intersecting arcs of equal radius from endpoints $A$ and $B$.