### Symbols

Negation, not

Conjunction, and

Inclusive or

Exclusive or

Conditional, if-then, implies

"If, and only if," logically equivalent

A tautology

A contradiction

For all

There exists

Such that

### Key Words

A statement that is taken to be true without argument.

An if-then statement.

The conjunction of two statements \(P\) and \(Q\) is the statement "\(P\) and \(Q\)".

A statement which is always false.

The contrapositive is a way to rewrite the conditional statement "if \(P\) then \(Q\)." The contrapositive is the statement "if not \(Q\) then not \(P\)." The conditional and contrapositive are logically equivalent.

Describes a fundamental relationship between \(\neg,\) \(\vee\) and \(\wedge.\)

The act of drawing a conclusion from a set of premises.

The disjunction of the two statements \(P\) and \(Q\) is the statement "\(P\) or \(Q\)". The disjunction refers to both inclusive and exclusive or.

An exclusive or means one or the other, but not both. The default in mathematics is the inclusive or. An exclusive or needs to be made explicit.

Two statements are logically equivalent. In other words, \(P\) if, and only if \(Q\) means either \(P\) and \(Q\) are both true or they are both false.

An inclusive or means one, the other, or both. This is the default or in mathematics. Also see exclusive or.

The negation of a statement \(P\) is to say "not \(P\)".

A statement which makes a contradiction true or a tautology false.

A function from a set \(S\) to the space of truth values \(\{T, F\}.\)

The universal quantifier is a statement about all element of a set. The existential quantifier is that there exists an element of a set with a certain property.

A sentence that is either true or false.

A statement which is always true.

An operations table that has a column for each input statement, a column for the result, and a row for each possible set of truth values of the input statements.