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

Negation

The negation of $A$ is denoted by $\neg A$. It asserts the opposite truth value to $A$. $\neg (\neg A)\equiv A$

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: $\text{For all integers }n\ge 5,2^n>n^2$

Two types of Quantified Statements

1. Universally Quantified statement is of the form

$\forall x\in S,P(x)$

2. Existentially quantified statement is of the form

$\exists x\in S,P(x)$

Negation of Quantifiers

$\neg (\forall x\in S,P(x))\equiv (\exists x\in S,\neg P(x))$ $\neg (\exists x\in S,P(x))\equiv (\forall x\in S,\neg P(x))$

Nested Quantifiers

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