Proposition

Proposition is a statement which is either true or false, e.g., " is irrational" or "". According to this definition, statements like is not a proposition (since the truth depends on the value of ).

Connectives: Propositions can be composed together via connectives (conjunction, disjunction, and negation).

• (Conjunction:) is true if and only if both and are true.

• (Disjunction:) is true if and only if at least one is true.

• (Negation:) is true if and only if is false.

• (Implication:) is false if and only if is true and is false. It is easy to check that is equivalent to . We encounter this kind of statement very often in mathematics, and there are several ways to say this statement, e.g,

  1. If , then
  2. if
  3. only if
  4. is sufficient for
  5. is necessary for
  6. implies
  7. is implied by

The converse of the proposition "" is written as , and the contrapositive is .

See the following table for more detail.