1.3 Predicates and Quantifiers

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

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

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

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

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

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

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

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

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

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

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

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

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

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