0. ICS 253: Discrete Structures I Predicates and Quantifiers
Spring Semester 2014 (2013-2)
Predicates and Quantifiers
Dr. Nasir Al-Darwish
Computer Science Department
King Fahd University of Petroleum and Minerals

1. Propositional Predicate
Definition: A propositional predicate P(x) is a statement that has a variable x.
Examples of P(x)
P(x) = “The Course x is difficult”
P(x) = “x+2 < 5”
Note: a propositional predicate is not a proposition because it depends on the value of x.
Example: If P(x) = “x > 3”, then P(4) is true but not P(1).
A propositional predicate is also called a propositional function

2. Propositional Predicate – cont.
It also possible to have more than one variable in one predicate, e.g., Q(x,y) =“x > y-2”
P(x) is a function (a mapping) that takes a value for x and produce either true or false.
Example: P(x) = “x2 > 2” , P: Some Domain  {T, F}
Domain of x is called the domain (universe) of discourse

3. Quantification
A predicate (propositional function) could be made a proposition by either assigning values to the variables or by quantification.
Predicate Calculus: Is the area of logic concerned with predicates and quantifiers.

4. Quantifiers
1. Universal quantifier: P(x) is true for all (every) x in the domain. We write x P(x)
2. Existential quantifier: there exists at least one x in the domain such that P(x) is true. We write x P(x)
3. Others: there exists a unique x such that P(x) is true. We write !x P(x)

5. Universal Quantification
Uses the universal quantifier  (for all)
x P(x) corresponds to “p(x) is true for all values of x (in some domain)”
Read it as “for all x p(x)” or “for every x p(x)”
Other expressions include “for each” , “all of”, “for arbitrary” , and “for any” (avoid this!)
A statement x P(x) is false if and only if p(x) is not always true (i.e., P(x) is false for at least one value of x)
An element for which p(x) is false is called a counterexample of x P(x); one counterexample is all we need to establish that x P(x) is false

6. Universal Quantification - Examples
Example 1: Let P(x) be the statement “x + 1 > x” . What is the truth value of ∀x P(x), where the domain for x consists of all real numbers?
Solution: Because P(x) is true for all real numbers x, the universal quantification ∀x P(x) is true.
Example 2: Suppose that P(x) is “x2 > 0” . What is the truth value of ∀x P(x), where the domain consists of all integers.
Solution: We show ∀x P(x) is false by a counterexample. We see that x = 0 is a counterexample because for x = 0, x2 = 0, thus there is some integer x for which P(x) is false.

7. Existential Quantification
Uses the existential quantifier  (there exists)
x P(x) corresponds to “There exists an element x (in some domain) such that p(x) is true”
In English, “there is”, “for at least one”, or “for some”
Read as “There is an x such that p(x)”, “There is at least one x such that p(x)”, or “For some x, p(x)”
A statement x P(x) is false if and only if “for all x, P(x) is false”

8. Existential Quantification - Examples
Example 1: Let P(x) denote the statement “x > 3”. What is the truth value of ∃x P(x), where the domain for x consists of all real numbers?
Solution: Because “x > 3” is true for some values of x , for example, x = 4, the existential quantification ∃x P(x) is true.
Example 2: Let Q(x) denote the statement “x = x + 1” . What is the truth value of ∃x Q(x), where the domain consists of all real numbers?
Solution: Because Q(x) is false for every real number x, the existential quantification ∃x Q(x) is false.

9. Predicates and Negations
When true
When false
Negation
x P(x)
For all x, P(x) is true
There is at least one x s.t. P(x) is false
x P(x)
x P(x)
There is at least one x s.t. P(x) is true
For all x, P(x) is false
x P(x)

10. Domain (or Universe) of Discourse
Cannot tell if a quantified predicate P(x) is true (or false) if the domain of x is not known.
The meaning of the quantified P(x) changes when we change the domain.
The domain must always be specified when universal or existential quantifiers are used; otherwise, the statement is ambiguous.

11. Quantification Examples
P(x) = “x+1 = 2” Domain is R (set of real numbers)
Proposition Truth Value
x P(x)
x  P(x)
x P(x)
x  P(x)
!x P(x)
!x  P(x) F F T T T F

12. Quantification Examples
P(x) = “x2 > 0”
Domain Proposition Truth Value
R x P(x)
Z x P(x)
Z - {0} x P(x)
Z !x  P(x)
N={1,2, ..} x  P(x) F F T T F

13. Quantification Examples
Proposition Truth Value
xR (x2  x)
!xR (x2 < x)
x(0,1) (x2 < x)
x{0,1} (x2 = x)
x P(x) F F T T T

14. Logically Equivalence
Definition: Two statements involving predicates & quantifiers are logically equivalent
if and only if
they have the same truth values independent of the domains and the predicates.
Examples:
 ( x P(x) )  x  P(x)
 ( x P(x) )  x  P(x)

15. x P(x)  x P(x)  Theorem P(x1)  P(x2)  ...  P(xn)
If the domain of discourse is finite, say Domain = {x1, x2, …, xn}, then
x P(x) 
x P(x) 
P(x1)  P(x2)  ...  P(xn)
P(x1)  P(x2)  ...  P(xn)

16. ((x P(x) )  ((!x Q(x) )  (x  P(x) )))
Precedence
 , , , , , 
 
Example:
((x P(x) )  ((!x Q(x) )  (x  P(x) )))

17. Correct Equivalences x ( P(x)  Q(x) )  x P(x)  x Q(x)
This says that we can distribute a universal quantifier over a conjunction
x ( P(x)  Q(x) )  x P(x)  x Q(x)
This says that we can distribute an existential quantifier over a disjunction
The preceding equivalences can be easily proven if we assume a finite domain for x = {x1, x2, …, xn}
Note: we cannot distribute a universal quantifier over a disjunction, nor can we distribute an existential quantifier over a conjunction.

18. Wrong Equivalences x 1 2 P(x) T F Q(x)
x ( P(x)  Q(x) )  x P(x)  x Q(x)
Read as “there exists an x for which both P(x) and Q(x) are true is equivalent to there exists an x for which P(x) is true and there exists an x for which Q(x) is true”. One can construct an example that makes the above equivalence false. Consider the following
x ( P(x)  Q(x) ) = F but
x P(x)  x Q(x) = T, since P(1) is true and Q(2) is true x 1 2
P(x) T F
Q(x)

19. Wrong Equivalences x 1 2 P(x) T F Q(x)
x ( P(x)  Q(x) )  x P(x)  x Q(x)
One can construct an example that makes the above equivalence false. Consider the following
x ( P(x)  Q(x) ) = T but
x P(x)  x Q(x) = F  F = F x 1 2
P(x) T F
Q(x)

20. Wrong Equivalences ( x P(x) )  Q(x)  x ( P(x)  Q(x) )
Notice that the LHS = ( x P(x) )  Q(x) is not fully quantified. So it cannot be equivalent to RHS.

21. Quantifiers with restricted domains
What do the following statements mean for the domain of real numbers?
Be careful about → and ˄ in these statements