1. 1.3 Predicates and Quantifiers

2. Predicates and Quantifiers
E.g., “If it is sunny, I’ll buy X.” Here the parameter is X.
Def: A predicate is a propositional function (a proposition with parameters).
Note:
When all parameters are assigned constant values, the predicate expression has one of the values T or F.
Another way to change a predicate into a proposition is to specify conditions on the parameters.
Chapter 1, section 3
Predicates and Quantifiers

3. Quantifiers Universal quantification Existential quantification
Def: The universal quantification of P(x) is the proposition “P(x) is true for all values of x in the universe of discourse.”
Chapter 1, section 3
Predicates and Quantifiers

4. Universal Quantification
Notation: ( x P(x))
This is read: “for all x P(x) is true.”
This could be used to express the concept:
“Every sunny day I buy a red bag.”
Notes: “==“ means “is equals to”
x (D(x)  C(x)) is translated
For all x, if D(x) then C(x)
x (C(x)  D(x)) is translated similarly
Chapter 1, section 3
Predicates and Quantifiers

5. Example, universal quantification
C(x) == "x has taken algebra” D(x) == "x is enrolled in discrete math”
x (D(x)  C(x)) {is True} but
x (C(x)  D(x)) {is False}
Chapter 1, section 3
Predicates and Quantifiers

6. Existential Quantification
Notation: ( x P(x))
This is read “there exists an x such that P(x) is true”
This could be used to express the concept that at least once I bought a red bag.
Def: Existential quantification of P(x) is the proposition “There exists an element x in the universe of discourse such that P(x) is true”
Chapter 1, section 3
Predicates and Quantifiers

7. Example, existential quantification
 x (C(x)  D(x))
This is TRUE
if we can find one person who has taken algebra
AND is enrolled in discrete math,
OR find one person who hasn't taken algebra.
Chapter 1, section 3
Predicates and Quantifiers

8. From English to logical expressions
Equivalent to :
for all
all
any for every
every for any any
for arbitrary
an arbitrary
for each
Equivalent to :
there exists
there is
there is at least one
there is some
for some
some
for at least one
Chapter 1, section 3
Predicates and Quantifiers

9. Predicates and Quantifiers
Binding Variables
E.g.,  y  x P (x,y,z)
bound variables y and x,
and free variable z.
Def: a quantifier binds or restricts the use of variables. Variables that are not bound are free.
Chapter 1, section 3
Predicates and Quantifiers

10. Predicates and Quantifiers
Consider:
z y x P(x,y,z)
for every possible value of z, there is at least one value of y such that for every possible value of x, P(x,y,z) is true
Chapter 1, section 3
Predicates and Quantifiers

11. Predicates and Quantifiers
More Examples
Suppose P(x,y,z) is the predicate
“When I teach discrete math in semester x, student y does well on exam z.”
Then x y z P(x,y,z) is the statement:
Every time I teach discrete math, there is at least one student who does well on every exam."
Suppose P(x,y, z) is the predicate
“When I teach discrete math in semester x, student y does well on exam z.”
Then x y z P(x,y,z) is the statement:
Every time I teach discrete math, there is at least one student who does well on every exam."
Chapter 1, section 3
Predicates and Quantifiers

12. Predicates and Quantifiers
Order matters!
y x z P(x,y,z)
would be true if "There is at least one student who, every time I teach discrete math, always does well on all my exams."
(why is he always taking the course?)
Chapter 1, section 3
Predicates and Quantifiers

13. Negation of quantifiers:
~ x P(x)   x ~P(x)
true:
P(x) is false for every x.
false:
There is an x for which P(x) is true.
~ x P(x)   x ~ P(x)
true:
There is an x for which P(x) is false.
false:
P(x) is true for every x.
~ x P(x) and  x ~ P(x)
~ x P(x) and  x ~P(x)
Chapter 1, section 3
Predicates and Quantifiers

14. Predicates and Quantifiers
Self Quiz
Simplify the following by moving “~” inside the quantifiers and connectors:
~  x  y  z ( P(x) V ( Q(y)  R(z)))
Chapter 1, section 3
Predicates and Quantifiers

15. Answer: ~  x  y  z ( P(x) V ( Q(y)  R(z)))
x  y  z (~ P(x)  (~ Q(y) V ~ R(z)))
Chapter 1, section 3
Predicates and Quantifiers