Rosen 1.3

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# Rosen 1.3 - PowerPoint PPT Presentation

Predicate Calculus. Rosen 1.3. Propositional Functions. Propositional functions (or predicates) are propositions that contain variables. Ex: Let P(x) denote x &gt; 3 P(x) has no truth value until the variable x is bound by either assigning it a value or by quantifying it. .

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Propositional 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 xP(x)
• True when P(x) is false for every x
• False if there is an x for which P(x) is true.
•  xP(x) is equivalent to xP(x)
• True if there exists 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: xU T(x,Bob)

 denotes membership

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.

x (x=Alice  T(Bob,x))

Bob trusts only Alice.x (x=Alice  T(Bob,x))

Let p be “x=Alice”

q be “Bob trusts x”

p q p  q

T T T

T F F

F T F

F F T

True only when Bob trusts Alice or Bob does not trust someone who is not Alice

Quantification of Two Variables(read left to right)
• xyP(x,y) or yxP(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) 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.
Quantification of Two Variables
• xyP(x,y)
• True when for every x there is a y for which P(x,y) is true.
• (in this case y can depend on x)
• False if there is an x such that P(x,y) is false for every y.
• yxP(x,y)
• True if there is a y for which P(x,y) is true for every x.
• (i.e., true for a particular y regardless (or independent) of x)
• False if for every y there is an x for which P(x,y) is false.
• Note that order matters here
• In particular, if yxP(x,y) is true, then xyP(x,y) is true.
• However, if xyP(x,y) is true, it is not necessary that yxP(x,y) is true.
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.

yxL(x,y)

Examples 3b1

There is somebody whom Lydia does not love.

xL(Lydia,x)

Nobody loves everybody. (For each person there is at least one person they do not love.)

xyL(x,y)

There is somebody (one or more) whom nobody loves

y x L(x,y)

Examples 3b2

There is exactly one person whom everybody loves.

xyL(y,x)?

No. There could be more than one person everybody loves

x{yL(y,x)  w[(yL(y,w))  w=x]}

If there are, say, two values x1 and x2 (or more) for which L(y,x) is true, the proposition is false.

x{yL(y,x)  w[(yL(y,w))  w=x]}?

xw[(y L(y,w))  w=x]?

Examples 3c

There are exactly two people whom Lynn loves.

x y{xy  L(Lynn,x)  L(Lynn,y)}?

No.

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.

xy(L(x,y)  x=y)

Thinking of Quantification as Loops

• Quantifications of more than one variable can be thought of as nested loops.
• For example, xyP(x,y) can be thought of as a loop over x, inside of which we loop over y (i.e., for each value of x).
• Likewise, xyP(x,y) can be thought of as a loop over x with a loop over y nested inside. This can be extended to any number of variables.

Quantification as Loops

• Using this procedure
• xyP(x,y) is true if P(x,y) is true for all values of x,y as we loop through y for each value of x.
• xyP(x,y) is true if P(x,y) is true for at least one set of values x,y as we loop through y for each value of x.
• …And so on….

Quantification of 3 Variables

• Let Q(x,y,z) be the statement “x + y = z”, where x,y,z are real numbers.
• What is the truth values of
• xyzQ(x,y,z)?
• zxyQ(x,y,z)?

Quantification of 3 Variables

• Let Q(x,y,z) be the statement “x + y = z”, where x,y,z are real numbers.
• xyzQ(x,y,z)
• is the statement, “For all real numbers x and for all real numbers y, there is a real number z such that
• x + y = z.”

True

Quantification of 3 Variables

Let Q(x,y,z) be the statement “x + y = z”, where x,y,z are real numbers.

zxyQ(x,y,z)

is the statement, “There is a real number z such that for all real numbers x and for all real numbers y,

x + y = z.”

False

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 byImplication 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 red”

and U is the set of all people.

No Georgia Tech students are ignorant.

x(P(x)  Q(x))

Also works. 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)) DeMorgan
• x (P(x) Q(x)) Implication equivalence
Examples 4b

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.

All ignorant people wear red.

x(Q(x) R(x))

Examples 4c

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 student wears red.

x(P(x) R(x))

x(R(x)  P(x))

Examples 4d

If “no Georgia Tech students are ignorant” and “all ignorant people wear red”, does it follow that “no Georgia Tech student wears red?”

x((P(x) Q(x))  (Q(x) R(x)))

NO

Some misguided GT student might wear red!!

This can be shown with a truth table or Wenn diagrams