Discreet Math - Chapter 2.1 - "Logical Form and Logical Equivalence " - FLASHCARDS

Vocabulary practice flashcards generated from lecture notes on logical form, statements, compound statements, truth values, logical equivalence, tautologies, and De Morgan's laws.

Argument

A sequence of statements aimed at demonstrating the truth of an assertion.

Conclusion

The assertion at the end of the sequence in an argument.

Premises

The preceding statements in an argument that aim to demonstrate the truth of the conclusion.

Statement (or Proposition)

A sentence that is true or false but not both.

Initial Undefined Terms in Logic

The foundational words sentence, true, and false, which remain undefined to avoid infinite regress in defining new logical terms.

Negation

represented as ¬p¬p or not p. not p, flips the truth value of a single statement. If the original statement pp is true, ¬p¬p will be false, and if pp is false, ¬p¬p will be true.

Conjunction

If $p$ and $q$ are statement variables, it is "$p$ and $q$", which means $p \land q$. It is true when, and only when, both $p$ and $q$ are true.

Disjunction

the logical operation where at least one of the statements is true. In our case, is represented as p or qp or q or mathematically as p→qp→q, written as p \/ q (not ×p×p). It is true if either pp is true, qq is true, or both are true, and only false when both are false.

Statement Form (or Propositional Form)

An expression made up of statement variables (such as $p$, $q$, and $r$) and logical connectives (such as ~ or $\sim$, $\land$, and $\lor$) that becomes a statement when actual statements are substituted for the component statement variables.

Logically Equivalent Statement Forms

Involving two statement forms if, and only if, they have identical truth values for each possible substitution of statements for their statement variables, denoted $P \equiv Q$.

Exclusive Or

A statement form representing "$p$ or $q$ but not both", symbolized as $(p \lor q) \land \sim(p \land q)$.

De Morgan's Laws

Logical equivalences stating that the negation of an and statement is logically equivalent to the or statement in which each component is negated ($\sim(p \land q) \equiv \sim p \lor \sim q$), and the negation of an or statement is logically equivalent to the and statement in which each component is negated ($\sim(p \lor q) \equiv \sim p \land \sim q$).

Tautology

A statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables.

Contradiction

A statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables.

Double Negative Law

The logical principle stating that the negation of the negation of a statement form is logically equivalent to the statement form itself ($\sim(\sim p) \equiv p$).

Neither-Nor Translation

translates symbolically to $\sim p \land \sim q$ ("not $p$ and not $q$").

But Translation

links two independent clauses and translates logically to "and" ($\land$).