1.3
Propositional Rosen Functions
Propositional functions (or predicates) are propositions that contain variables. Ex: Let P(x) denote x > 3 P(x) has no truth value until the variable x is bound by either
assigning it a value or by quantifying it.
Assignment of values
Let Q(x,y) denote "x + y = 7". Each of the following can be determined as T or F. Q(4,3) Q(3,2) Q(4,3) Q(3,2) ~[Q(4,3) Q(3,2)]
Quantifiers
Universe of Discourse, U: The domain of a variable in a propositional function. Universal Quantification of P(x) is the proposition:"P(x) is true for all values of x in U." Existential Quantification of P(x) is the proposition: "There exists an element, x, in U such that P(x) is true."
Universal Quantification of P(x)
xP(x) "for all x P(x)" "for every x P(x)" Defined as: P(x0) P(x1) P(x2) P(x3) . . . for all xi in U Example: Let P(x) denote x2 x If U is x such that 0 < x < 1 then xP(x) is false. If U is x such that 1 < x then xP(x) is true.
Existential Quantification of P(x)
xP(x) "there is an x such that P(x)" "there is at least one x such that P(x)" "there exists at least one x such that P(x)" Defined as: P(x0) P(x1) P(x2) P(x3) . . . for all xi in U Example: Let P(x) denote x2 x If U is x such that 0 < x 1 then xP(x) is true. If U is x such that x < 1 then xP(x) is true.
Quantifiers
xP(x) True when P(x) is true for every x. False if there is an x for which P(x) is false. xP(x) True if there exists an x for which P(x) is true. False if P(x) is false for every x.
Negation (it is not the case)
xP(x) equivalent to xP(x) True when P(x) is false for every x False when there is an x for which P(x) is true. xP(x) is equivalent to xP(x) True is there is an x for which P(x) is false. False if P(x) is true for every x.
Examples 2a
Let T(a,b) denote the propositional function "a trusts b." Let U be the set of all people in the world. Everybody trusts Bob. xT(x,Bob) Could also say: xU T(x,Bob) Bob trusts somebody. xT(Bob,x)
Examples 2b
Alice trusts herself. T(Alice, Alice) Alice trusts nobody. x T(Alice,x) Carol trusts everyone trusted by David. x(T(David,x) T(Carol,x)) Everyone trusts somebody. x y T(x,y)
Examples 2c
x y T(x,y) Someone trusts everybody. y x T(x,y) Somebody is trusted by everybody. Bob trusts only Alice. T(Bob, Alice) x (x=Alice T(Bob,x))
Bob trusts only Alice. T(Bob, Alice) x (x=Alice T(Bob,x)) Let p be "x=Alice" q be "Bob trusts x" p T T F F q T F T F p q T T F T
False only when Bob trusts someone who is not Alice
Quantification of Two Variables
(read left to right)
xyP(x,y) or yxP(x,y) True when P(x,y) is true for every pair x,y. False if there is a pair x,y for which P(x,y) is false. x yP(x,y) True when for every x there is a y for which P(x,y) is true. False if there is an x such that P(x,y) is false for every y.
Quantification Two of Variables
xyP(x,y) True if there is an x for which P(x,y) is true for every y. False if for every x there is a y for which P(x,y0 is false. x yP(x,y) or y xP(x,y) True if there is a pair x,y for which P(x,y) is true. False if P(x,y) is false for every pair x,y.
Examples 3a
Let L(x,y) be the statement "x loves y" where U for both x and y is the set of all people in the world. Everybody loves Jerry. xL(x,Jerry) Everybody loves somebody. x yL(x,y) There is somebody whom everybody loves. yxL(x,y)
Examples 3b
Nobody loves everybody. x yL(x,y) There is somebody whom Lydia does not love. xL(Lydia,x) There is somebody whom no one loves. xyL(y,x) There is exactly one person whom everybody loves. x{yL(y,x) z[(wL(w,z)) z=x]}
Examples 3c
There are exactly two people whom Lynn loves. x y{x y L(Lynn,x) L(Lynn,y) z[L(Lynn,z) (z=x z=y)]} Everyone loves himself or herself. xL(x,x) There is someone who loves no one besides himself or herself. xy(L(x,y) x=y)
Examples 4a
Let P(x) be the statement: "x is a Georgia Tech student" Q(x) be the statement: " x is ignorant" R(x) be the statement: "x wears red" and U is the set of all people. No Georgia Tech students are ignorant. x(P(x) Q(x)) x(P(x) Q(x)) OK by Implication equivalence. x(P(x) Q(x)) Does not work. Why?
Examples 4a
No Georgia Tech students are ignorant. x(P(x) Q(x)) x(P(x) Q(x)) x (P(x) Q(x)) Negation equivalence x ( P(x) Q(x)) Implication equivalence x ( P(x) Q(x))DeMorgans x ( P(x) Q(x)) Double negation Only true if everyone is a GT student and is not ignorant.
Examples 4a
P(x) be the statement: "x is a Georgia Tech student" Q(x) be the statement: " x is ignorant" R(x) be the statement: "x wears...