Propositional Functions Rosen (6 th Ed.) 1.3, 1.4 Propositional functions (or predicates) are propositions that contain variables. Ex: 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 Q(x,y) denote x + y = 7. Each of the following can be determined as 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: he 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: here 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(x 0 ) P(x 1 ) P(x 2 ) P(x 3 )... for all x i in U Example: (½) 2 < ½ P(x) denote x 2 x If U is x such that 0 < x 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(x 0 ) P(x 1 ) P(x 2 ) P(x 3 )... for all x i in U Example: 1 2 1 P(x) denote x 2 x If U is x such that 0 < x < 1 then xp(x) is false. If U is x such that 0 < x 1 then xp(x) is true. 1
Quantifiers xp(x) rue when P(x) is true for every x. False if there is an x for which P(x) is false. xp(x) rue if there exists an x for which P(x) is true. False if P(x) is false for every x. Precedence of Quantifiers and have higher precedence than all other logical operators (,,, etc.). x P(x) Q(x) means ( x P(x)) Q(x) x P(x) Q(x) does not mean x (P(x) Q(x) ) Negation (it is not the case) xp(x) is equivalent to x P(x) rue when P(x) is false for every x False when there is an x for which P(x) is true. Example: P(x) be the statement x is a USC fan and the Universe of Discourse be the students in our CPSC2070 class. xp(x) It is not the case that there is a student in CPSC2070 who is a USC fan. x P(x) For all students in CPSC2070 it is not the case that one of us is a USC fan. Negation (it is not the case) xp(x) is equivalent to x P(x) rue is there is an x for which P(x) is false. False if P(x) is true for every x. Example: P(x) be the statement x is a Clemson fan and the Universe of Discourse be the students in our CPSC2070 class. xp(x) It is not the case every student in CPSC2070 is a Clemson fan. x P(x) here exists a student in CPSC2070 who is not a Clemson fan. Examples (a,b) denote the propositional function a trusts b. U be the set of all people in the world. Everybody trusts Bob. x(x,bob) Could also say: x U (x,bob) Bob trusts somebody. x(bob,x) Alice trusts herself. (Alice, Alice) Alice trusts nobody. x (Alice,x) Examples Carol trusts everyone trusted by David. x((david,x) (Carol,x)) Bob trusts only Alice. (Bob, Alice) x (x=alice (Bob,x)) 2
Bob trusts only Alice. (Bob, Alice) x (x=alice (Bob,x)) p be x=alice q be Bob trusts x p q p q F F F F F False only when Bob trusts someone who is not Alice Quantification of wo Variables (read left to right) x yp(x,y) or y xp(x,y) rue 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) or y xp(x,y) rue 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. Quantification of wo Variables x yp(x,y) rue 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. x yp(x,y) rue 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,y) is false. P(x,y) be the statement x+y = 7 Is x yp(x,y) true or false? For every number x we can find a number y such that x + y = 7. Is x yp(x,y) true or false? For every number x we can find a number y such that x + y 7. Examples 3a 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) here is somebody whom everybody loves. y xl(x,y) Nobody loves everybody. x y L(x,y) Examples 3b here is somebody whom Lydia does not love. x L(Lydia,x) here is somebody whom no one loves. x y L(y,x) here is exactly one person whom everybody loves. x{ yl(y,x) z[( wl(w,z)) z=x]} 3
Examples 3c here 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) here is someone who loves no one besides himself or herself. x y(l(x,y) x=y) P Q Q P(x) Q(x) F F F x( P(x) Q(x)) F F OK by Implication equivalence. F F x(p(x) Q(x)) Does not work. Why? x(p(x) Q(x)) x (P(x) Q(x)) DeMorgans for quantifiers x ( P(x) Q(x)) Implication equivalence x ( P(x) Q(x)) DeMorgans x ( P(x) Q(x)) Double negation x ( P(x) Q(x)) P Q Q P(x) Q(x) F F F F F F F F F Only true if everyone is a Clemson student and is not ignorant. OK x( P(x) Q(x)) Implication equivalence. x(p(x) Q(x)) Was not equivalent x(p(x) Q(x)) Is equivalent. Why? x(p(x) Q(x)) x (P(x) Q(x)) DeMorgan for quantifyers x ( P(x) Q(x)) DeMorgan x (P(x) Q(x)) Implication equivalence 4
Examples 4b R Q R(x) Q(x) F F F F All ignorant people wear red. x(r(x) Q(x)) Q R Q(x) R(x) Is this right? No! his says there may be some ignorant people wearing orange! x(q(x) R(x)) F F F F F F Examples 4c No Clemson student wears red. x(p(x) R(x)) x(r(x) P(x)) Both are correct since one is the contraposition of the other. 5