# Mathematical Statement

Definition 1.3.1

A

statementis a sentence that has a definite state of being either true or false.

Statement = Quantifier + Open Sentence → closed sentence

Examples $1.2+1=6$ (A false statement.) $2.π+2>5$ (A true statement.)

### Negation

The negation of $A$ is denoted by $¬A$. It asserts the opposite truth value to $A$. $¬(¬A)≡A$

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)

Ex: $For all integersn≥5,2_{n}>n_{2}$

#### Two types of Quantified Statements

- Universally Quantified statement is of the form

$∀x∈S,P(x)$

- Existentially quantified statement is of the form

$∃x∈S,P(x)$

#### Negation of Quantifiers

$¬(∀x∈S,P(x))≡(∃x∈S,¬P(x))$ $¬(∃x∈S,P(x))≡(∀x∈S,¬P(x))$

#### Nested Quantifiers

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