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Given the statements p and q, an implication is a statement that is false when p is true and q is false, and true otherwise.

↔

A biconditional statement is true whenever the truth value is the same for both p and q and false otherwise.

¬

'NOT:'Negation - a method of assigning the opposite truth value to the statement.

∧

'AND:' Given p and q, a conjunction is the proposition that is true when both p and q are true and is false otherwise.

∨

'OR:' Given p and q, a disjunction is the proposition that is false when both p and q are false, but is true otherwise.

⊕

'XOR:' An exclusive or is a proposition which is true when exactly one of p and q is true and is false otherwise.

∀

The universal quantification of P(x) is the proposition "P(x) is true for all values x in the universe of discourse."

∃

The existential quantification of P(x) is the proposition "There exists an element x in the universe of discourse such that P(x) is true."