RMIT University; Taylor's College This is a story about four people named Everybody, Somebody, Anybody and Nobody. There was an important job to be done

RMIT University; Taylor's College

This is a story about four people named Everybody, Somebody, Anybody and Nobody.

There was an important job to be done and Everybody was sure that Somebody would do it. Anybody could have done it, but Nobody did it. Somebody got angry about that, because it was Everybody's job. Everybody thought that Anybody could do it, but Nobody realized that Everybody wouldn't do it.

It ended up that Everybody blamed Somebody when Nobody did what Anybody could have done.


Lecture 8

To apply quantifiers on predicatesTo apply the de Generalized de Morgan’s LawsTo determine the truth value of predicates involving combination of two quantifiers

http://upload.wikimedia.org/wikipedia/commons/2/2c/De_Morgan_Augustus.jpg


Universal Quantifiers: All

We may says that: “Every student has brains.”

Let be a student. Let D = All MATH 2111 students.

This is a set – the domain of interpretation.

So “every student has brains” becomes

brains has , xDx


Universal Quantifiers Let P(x) represent the predicate: “Student x has

brains.” Then “every student has brains” becomes:

or more simply,

When we don’t need to specify the domain, this becomes:

trueis )P( , xDx

)P( , xx

)P( , xDx


Let stand for a tall building more than 10 floors, and

let is made of steel and concrete. Use the Universal Quantifier to write a

predicate. Write also, the meaning of the predicate.

Universal QuantifiersExercise:


Universal QuantifiersExercise:

For all tall building more than 10 floors (, the building is made of steel and concrete ( is true).


Existential Quantifiers:There Exist How do we say, “There exists a student with

brains”?

How do we say, “Some students have brains”?

In logic, “some” means “at least one”.

This is how we apply one of the quantifiers to a predicate.

)P( , xx

)P( , xx

,


Let stand for a Deluxe suite in a resort, and

let is empty. Use the Existential Quantifier to write

a predicate. Write also, the meaning of the predicate.

Existential QuantifiersExercise:


Existential Quantifiers:There Exist

There exist a deluxe suite in a resort () such that, the resort is empty (is true ).


Quantifiers A statement involving predicates whose

variables are all properly quantified becomes a proposition (provided that the domain is known).


Quantifiers Example: D = Z = {…, -3, -2, -1, 0, 1, 2, 3, …}

P(x) means “x is positive” Which are true?

This means “for all x, P(x)” which means “for all x, x is positive.” This is false. A counterexample is x = 0 (or x = -1, etc) A counterexample is an example which proves that a

universally quantified statement is false.

This means “there exists x such that P(x) is true” which means “there exists x such that x is positive”. This is true. For example, x = 1.

)P( ,or )P( , xxxx

)P( , xx

)P( , xx


Quantifiers It’s more likely that

is true than that

is true.

In fact, provided that the domain D is nonempty,

)(xxP

)(xxP

)()( xxPxxP


The Generalized de Morgan Laws What happens if we negate an expression involving

predicates and quantifiers?

The Generalized de Morgan Laws

Examples:1. “It’s not true that all food is delicious” is the same as

“there exists some food which is not delicious.”2. “It’s not true that some dogs bite” is the same as “there

aren’t any dogs who bite” or equivalently “all dogs don’t bite”.

)()]([ xPxxxP

)()]([ xPxxxP


Combining Quantifiers A predicate can have numerous variables, each

of which may be quantified. Example

It can be difficult to interpret expressions involving 3 or more quantifiers.

),,( zyxPzyx

),,(~)],,( [~ zyxPzyxzyxPzyx Negation


Combination of Quantifiers Having two quantifiers is a lot easier.

What do these mean? Which are true in any given situation?

This depends on how P(x, y) is defined, and on what set is chosen as the domain of interpretation D.

),( yxPyx ),( yxPyx),( yxPyx),( yxPyx

),( yxPxy

),( yxPxy


http://newsimg.bbc.co.uk/media/images/46343000/gif/_46343078_height_world_leaders_466.gifAccessed 29th September 2009

P(x, y): x is taller than y

Everyone is taller than everyone.

There exists a person who is taller than everyone.

Everyone is taller than someone.

Everyone is shorter than someone.

There is someone who is shorter than everyone.

There is someone who is taller than another person.

http://newsimg.bbc.co.uk/media/images/46343000/gif/_46343078_height_world_leaders_466.gif


Example Suppose P(x, y) means x ≥ y. Let D be the set

N \ {0} = {1, 2, 3, …}.

Which of the six predicate formulae given in the previous slide are true?

The discussion is in the following slides.


For all x and (for all) y, P(x, y) is true.

This says that no matter which numbers x and y we choose from N \ {0}, it will always happen that x ≥ y.

Is this true?

No. For example, x = 1 and y = 2.

),( yxPyx


For all x there exists y such that x ≥ y.

Here y can depend on x.

A different choice of x may lead to a different value of y.

Is this true?

Yes. For example, given x we can take y = x. Then x ≥ y. Or, we could take y = 1 when x = 1, and take y = x – 1 for

all other values of x. Or we could take y = 1 always.

),( yxPyx


There exists x such that for all y, x ≥ y.

This says that x is a constant, and every choice of y makes x ≥ y.

Is this true?

No. There is no such constant. (It would have to be the biggest integer – the largest element of N \ {0}. But this set has no largest element.)

),( yxPyx


There exists x and there exists y such that x ≥ y.

Is this true?

Yes. For example, take x = 2 and y = 1.

),( yxPyx


For all y there exists x such that x ≥ y.

This says that for every choice of y it’s possible to find an x which is ≥ y.

Is this true?

Yes. For example, put x = y + 1 (or take x = y)

),( yxPxy


There exists y such that for all x, x ≥ y.

This says that there is a constant y which is less than or equal to all values of x.

Is this true?

Yes: y = 1 has this property. It’s the smallest element of the set.

),( yxPxy


Exercises1. How do the results change if D changes to the finite set

{1, 2, 3, 4, 5}?

2. How do the results change if D changes from N \ {0} to Z?

3. Is there a domain for which all six formulae are true?

4. How do the results change if P(x, y) changes to “x > y”?

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RMIT University; Taylor's College This is a story about four people named Everybody, Somebody, Anybody and Nobody. There was an important job to be done