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Rosen 1.3

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 xP(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 xP(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: xU 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)

- 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.
- xyP(x,y) or yxP(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

- xyP(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.
- yxP(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 yxP(x,y) is true, then xyP(x,y) is true.
- However, if xyP(x,y) is true, it is not necessary that yxP(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.

yxL(x,y)

Examples 3b1

There is somebody whom Lydia does not love.

xL(Lydia,x)

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

xyL(x,y)

There is somebody (one or more) whom nobody loves

y x L(x,y)

There is exactly one person whom everybody loves.

xyL(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]}?

xw[(y L(y,w)) w=x]?

Examples 3c

There are exactly two people whom Lynn loves.

x y{xy L(Lynn,x) L(Lynn,y)}?

No.

x y{xy 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)

Thinking of Quantification as Loops

- Quantifications of more than one variable can be thought of as nested loops.
- For example, xyP(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, xyP(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.

- Using this procedure
- xyP(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.
- xyP(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….

- Let Q(x,y,z) be the statement “x + y = z”, where x,y,z are real numbers.
- What is the truth values of
- xyzQ(x,y,z)?
- zxyQ(x,y,z)?

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

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

zxyQ(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))

What about this?

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

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