Discrete Vocab. - Logic

Implications

(also known as implies, or If ... then) is a logical operation. The statement "p implies q" means that if p is true, then q must also be true. The statement "p implies q" is also written "if p then q" or sometimes "q if p." Statement p is called the premise of the implication and q is called the conclusion.

Statement

A statement (or proposition) is a sentence that is either true or false (both not both). So '3 is an odd integer' is a statement.

Self-Contradictions

A statement is self-contradictory if it is logically false, that is, if it is logically impossible for the statement to be true.

And

Given statements P and Q, we can combine them with various connectives. P and Q is true only when both P and Q are both true.

Equivalent Compound Statements

statements using two or more logic operations.

Conditional Statement

symbolized by p q, is an if-then statement in which p is a hypothesis and q is a conclusion

Inclusive or

the connective that gives the value true to a disjunction if either or both of the disjuncts are true.

Converse

the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P

Negation

the opposite of a given mathematical statement. One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true).

Negation of a Conditional Statement

to find a conditional statement to be false. The negation of a true statement will be false, and the negation of a false statement will be true.

If...Then

A conditional statement; imposing, depending on, or containing a condition

Or

P or Q is true if one or the other or both P and Q is true.

DeMorgan's Law

The complement of the union of two sets is equal to the intersection of the complements of the sets. The complement of the intersection of two sets is equal to the union of the complements of the sets.

Biconditional

a compound statement formed by combining two conditionals under "and." Biconditionals are true when both statements (facts) have the exact same truth value.

Disjunction

a compound statement formed by joining two statements with the connector "or".

Conjunction

a compound statement formed by joining two statements with the connector AND. The conjunction "p and q" is symbolized by p q. A conjunction is true when both of its combined parts are true; otherwise it is false.

Truth Table

a diagram in rows and columns showing how the truth or falsity of a proposition varies with that of its components. Creating a truth table is a systematic way of determining when a compound statement is true and when it is false.

Euler's Diagrams

a diagrammatic means of representing sets and their relationships (Venn diagram)

Simple Statement

a direct statement p: You are absent q: You have a make up assignment to complete.

Exclusive or

a logical operation that outputs true only when both inputs differ (one is true, the other is false)

Quantified Statement

a simple statement in predicate logic whose subject is qualified by either the universal quantifier or the existential quantifier. That is, it is either a universal statement or an existential statement.

Compound Statement

a statement that holds conditions p q : If you are absent, then you have a make up assignment to complete.

Tautology

a statement that is true by necessity or by virtue of its logical form.

Antecedent

a thing or event that existed before or logically precedes another.

Connectives

a word or phrase whose function is to link linguistic units together.

Quantifers

an expression (e.g., all, some ) that indicates the scope of a term to which it is attached.

Some

an unspecified amount or number of.

Contrapositive

are conditional statements that are written differently, but hold the same logical equivalence

None

by no amount; not at all

Consequent

following as a result or effect.

Conditional

is a compound statement formed by combining two sentences (or facts) using the words "if ... then." Subject to one or more conditions or requirements being met; made or granted on certain terms.

Inverse

type of conditional sentence which is an immediate inference made from another conditional sentence. Any conditional sentence has an inverse: the contrapositive of the converse

No

used to indicate that something is quite the opposite of what is being specified.

If and Only If

used to introduce a condition that is necessary as well as sufficient.

All

used to refer to the whole quantity or extent of a particular group or thing.