Mathematical Statement

Definition 1.3.1

A statement is a sentence that has a definite state of being either true or false.

Statement = Quantifier + Open Sentence closed sentence

Examples (A false statement.) (A true statement.)


The negation of is denoted by . It asserts the opposite truth value to .

If two things have the same truth values, we say that they are logically equivalent.

Quantified Statement

A quantified statement contains four parts:

  • a quantifier (universal, or existential);
  • a variable;
  • a domain (any set);
  • an open sentence involving the variable (that is either true or false whenever a value of the variable chosen from the domain is specified)


Two types of Quantified Statements

  1. Universally Quantified statement is of the form

  1. Existentially quantified statement is of the form

Negation of Quantifiers

Nested Quantifiers

two universal quantifiers or two existential quantifiers result in both statements have the same truth values.