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Presentation on theme: "Chapter 3 The Logic of Quantified Statements"— Presentation transcript:

1 Chapter 3 The Logic of Quantified Statements

2 Quantified Statements

A quantified statement is a statement that specifies quantity (of something).

Examples:

There is at least one student absent today.

Everybody in the class passes the 1st quiz.

The equation x2 + 5x – 6 = 0 has exactly two different real solutions.

A non-quantified statement: “3331 is a prime number.”

3 Quantified Statements

Most interesting properties (theorems) in mathematics are quantified statements.

Examples:

All polynomial functions are differentiable.

For all real number x, sin2x + cos2x = 1.

For any whole number n > 1, there is always a prime number between n and 2n.

4 Quantifiers

A quantifier is a symbols that expresses the quantity of a certain type of objects.

Only two types of quantifiers will be needed.

Universal quantifier: 

x(x2  0) means “for all x, x2  0”

Existential quantifier: 

n (n×n = n+n) means “there exists an n such that n×n = n+n”

5 Quantifiers

How do we express “there exists only one” with a quantified statement ?

! x P(x) means there is exactly one x such that P(x) is true.

It is the abbrev. of

 x { P(x)  [y (y  x  P(y))]}

Step-by-step explanation: