1. Discrete Mathematics Lecture 3: Predicate and Quantifier
By:
Nur Uddin, Ph.D

2. Review the previous lecture
Proposition symbol

3. Review the previous lecture
Truth tables

4. Review the previous lecture
Truth tables

5. Compound proposition
Example:
Solution:

6. Compound proposition

7. Logical equivalence

8. Laws of logic

9. Predicate logic

10. Motivation
Propositional logic provides a useful setting in which we can analyse many types of logical argument. There are situations, however, where propositional logic is inadequate, because it cannot deal with the logical structure that is sometimes present within atomic propositions.
Examples:
‘All even numbers are integers. 8 is an even number. Therefore 8 is an integer.’
‘It is not true that all prime numbers are odd. Therefore there must be at least one prime number that is not odd.’

11. Motivation (cont’d) From the example
‘All even numbers are integers’, ‘8 is an even number’, and ‘8 is an integer’
It has atomic proposition as follows:
All even numbers are integers’,
‘8 is an even number’,
‘8 is an integer’
The argument takes the following form:
In order to be able to analyse arguments such as these, we need to look at the logical structure within atomic propositions. Predicate logic allows us to do this.

12. Predicate
A predicate is a statement containing one or more variables. If values are assigned to all the variables in a predicate, the resulting statement is a proposition.

13. Predicate The statement “x is greater than 3” has two parts:
The first part, the variable x, is the subject of the statement.
The second part—the predicate, “is greater than 3”— refers to a property that the subject of the statement can have.
We can denote the statement “x is greater than 3” by P(x), where P denotes the predicate “is greater than 3” and x is the variable.
The statement P(x) is also said to be the value of the propositional function P at x.
Once a value has been assigned to the variable x, the statement P(x) becomes a proposition and has a truth value

14. Example 1
Let P(x) denote the statement “x > 3.” What are the truth values of P(4) and P(2)?
Let Q(x, y) denote the statement “x = y + 3.” What are the truth values of the propositions Q(1, 2) and Q(3, 0)?

15. Example 2
“All even numbers are integers. 8 is an even number. Therefore 8 is an integer.”
The atomic propositions are:
‘All even numbers are integers’,
‘8 is an even number’,
‘8 is an integer’
The argument takes the following form:

16. Example 3
“It is not true that all prime numbers are odd. Therefore there must be at least one prime number that is not odd”
The atomic propositions:
‘All prime numbers are odd’
‘There must be at least one prime number that is not odd’
The argument takes the form:

17. Quantifiers
When the variables in a propositional function are assigned values, the resulting statement becomes a proposition with a certain truth value.
However, there is another important way, called quantification, to create a proposition from a propositional function. Quantification expresses the extent to which a predicate is true over a range of elements.
In English, the words all, some, many, none, and few are used in quantifications.

18. Predicate and quantifiers in symbolic
A predicate P that contains a variable x can be written symbolically as P(x).
In general, a predicate with n variables, x1, x2, ..., xn, can be written P(x1, x2, ..., xn).
The quantifiers:
: for all
 : there exists

19. Example 1

20. Example 2
Solution:
or:

21. Mahasiswa INF 2017 usianya kurang dari 20 dan lebih dari 20.

22. Example 3
Solution:

23. Quantification types Two types of quantification:
Universal quantification  tell us that a predicate is true for every element under consideration
existential quantification  tells us that there is one or more element under consideration for which the predicate is true.
The area of logic that deals with predicates and quantifiers is called the predicate calculus.

24. Universal Quantifier
Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the domain of discourse (or the universe of discourse), often just referred to as the domain.

25. Universal Quantifier
Example:

26. Universal Quantifier
Example:
Example:

27. Existensial Quantifier
Besides the phrase “there exists,”we can also express existential quantification in many otherways, such as by using the words “for some,” “for at least one,” or “there is.” The existential quantification ∃xP(x) is read as:
“There is an x such that P(x),”
“There is at least one x such that P(x),”
“For some xP(x).”

28. Existensial Quantifier
Example 1:
Example 2:

29. The Uniqueness Quantifier
The uniqueness quantifier, denoted by ∃! or ∃1.
The notation ∃!xP(x) or [∃1xP(x)] states “There exists a unique x such that P(x) is true.”
For example, ∃!x(x − 1 = 0), where the domain is the set of real numbers, states that there is a unique real number x such that x−1=0. This is a true statement, as x = 1 is the unique real number such that x − 1 = 0.

30. Homework 1 1. 2.