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OutlineFirst Order Predicate Calculus

From English to Quantified StatementWorking with Quantifiers

Summary

Quantifiers

Alice E. Fischer

CSCI 1166 Discrete Mathematics for ComputingFebruary, 2018

Alice E. Fischer Quantifiers. . . 1/34



Summary

1 First Order Predicate CalculusPredicates and their Truth SetsSets of Numbers

2 From English to Quantified StatementUniversal QuantifiersExistential QuantifiersPractice

3 Working with QuantifiersNegating Quantified StatementsPracticeProofs

4 Summary




Summary

Predicates and their Truth SetsSets of Numbers

First Order Predicate Calculus

Predicates and their Truth Sets

Sets of Numbers




Summary


Predicates

In English, a sentence has a subject (noun or pronoun) and apredicate (verb phrase).

In logic, a proposition is a sentence that can be true or falsebut not both.

We can write a proposition using symbols – but we assignspecific meanings to those symbols. Often, the propositionmodels some real-world situation.

If the subject of a sentence is a variable, it is not a proposition.

We call it a predicate or an open sentence.

The domain of the predicate is the set of all values that canbe substituted for the variable.




Summary


The Truth Set of a Predicate

The truth set of a predicate, P, is the set of all values, x , in itsdomain, D, that produce true propositions when substituted forthe predicate’s variable. {x ∈ D | P(x)}Suppose our domain is R, the real numbers. Let P be thepredicate x2 > x . What is its truth set?

P is true for all values > 1

P is false for values −1 . . . 1, including both end points.

P is true for values < −1 (their squares are positive).

We can diagram this truth set using a number line:..........−2..........−1..........0..........1..........2..........< ........true........)(.......false.......)(...true......... >




Summary


Names for Sets

In mathematical work, some sets are used so often that someonegave them short names:

R: The set of all real numbers.

Z : The set of all integers

Q: The set of all rational numbers (quotients)

Add a superscript + to restrict the set to positive numbers.

Add a superscript − to restrict the set to negative numbers.

Use superscript nonneg for positives plus zero.




Summary

Universal QuantifiersExistential QuantifiersPractice

From English to Quantified Statement

Universal Conditional Statements

Practice




Summary


First Order Predicate Calculus

In the propositional calculus we have propositions withoutvariables.

In the predicate calculus we have predicates containing variables.

In the first order predicate calculus we quantify over variables.

In the second order predicate calculus, we quantify over sets ofvariables and/or over uninterpreted predicate symbols, but that isway beyond the scope of this course.




Summary


The Universal Quantifier

We often make universal statements (at the risk of soundingprejudiced):

All stars are a long long way from Earth.

These can be symbolized using the universal quantifier, ∀

Let S be the set of all stars that are celestial bodies. (Excludemovie stars, sports stars, paper stars, etc.)

Let W be the predicate “y is a long way from Earth.”

We can write:

∀y ∈ S ,W (y)

The predicate starts with a quantifier, a variable name, and thedomain of that variable and ends with an assertion.




Summary


The Existential Quantifier

We often make claims such as:

Somebody out there likes me.

These can be symbolized using the existential quantifier, ∃

Let P be the set of all people.

Let L be the predicate “x likes me.”

We can write:

∃x ∈ P, L(x)




Summary


Universal vs. Existential

A universally quantified statement is true if and only if thepredicate is true for every element of the domain.

Common terms that correspond to universal quantification include:for all, for every, for arbitrary, for any, for each, and given any.

An existentially quantified statement is true if the predicate is truefor even one element of the domain.

Common terms that correspond to existential quantificationinclude: there exists, there is a, we can find a, there is at least one,for some, and for at least one.




Summary


Practice: From English to a Quantified Statement

Start with this sentence:

All fish die when removed from the water.

Define a symbol for the predicate.

What is the domain of your predicate?

Symbolize the statement using the universal quantifier.




Summary


Practice

All fish die when removed from the water.

Let D be “f dies when removed from the water”.

The domain of D is L: living fish swimming in the water.

∀f ∈ L,D(f )




Summary


Universal Conditional Statements

The universal conditional statement is a generalization of theconditional or implication statement in propositional calculus:P → Q.

∀x , if P(x) then Q(x) or

∀x ,P(x)→ Q(x).

This might be written P(x)⇒ Q(x), meaning that every elementx that makes P true makes Q true.

There is no existential conditional.




Summary


Implicit Conditional Statements

Vegans do not eat products derived from animal sources.V= vegans A = x eats animal products∀x ∈ V ,∼ A(x) or

P = all people V = x is vegan A = x eats animal products∀x ∈ P,V (x)→ ∼ A(x)

All that glitters is not gold.T= glittery objects G = x is gold∀x ∈ T ,∼ G (x) or

O = all objects T = x glitters G = x is gold∀x ∈ O,T (x)→ ∼ G (x)




Summary



Translate these statements into quantified predicates:

1 All UNH students have a student ID number.

2 Some UNH Engineering students are CS majors.

3 A student must work hard to graduate in Engineering.




Summary



1 All UNH students have a student ID number.2 ∀x ∈ UNH students, ID(x) or∀x ∈ students,UNH(x)→ ID(x)

1 Some UNH Engineering students are CS majors.2 ∃ z ∈ UNH Engineering students,CS(z) or∃ z ∈ students,UNH Engineering(z) ∧ CS(z)

1 A student must work hard to graduate in Engineering.2 ∀ s ∈ Engineering students,Graduate(s)→WorksHard(s) or

∀ s ∈ students,UNH Engineering(s) ∧ Graduate(s)→WorksHard(s)




Summary

Negating Quantified StatementsPracticeProofs

Working with Quantifiers





Summary


Negating Quantified Statements

The negation of a universally quantified statement is a negativeexistential statement.

∼ ∀x ∈ P, L(x) is equivalent to

∃x ∈ P,∼ L(x)

This follows from the fact that ∀x ∈ P, L(x) really meansL(x1) ∧ L(x2) ∧ L(x3)....

And then ∼ (L(x1) ∧ L(x2) ∧ L(x3)...) is∼ L(x1)∨ ∼ L(x2)∨ ∼ L(x3)...,

which is ∃x ∈ P,∼ L(x).




Summary


Negating Quantified Statements

The negation of a existentially quantified statement is a negativeuniversal statement.

∼ ∃x ∈ P, L(x) is equivalent to

∀x ∈ P,∼ L(x)

Why is this true?




Summary


Negating an Implication

Implications and their negations are formulas with specialimportance in logic. The negation follows from all the previousrules. Here we develop the solution step by step.

∼ (∀x ,P(x)→ Q(x))

∃x ,∼ (P(x)→ Q(x))

∃x ,P(x) ∧ ∼ Q(x)




Summary


Practice: Negating Quantified Statements

Remember:

1 The negation of a universally quantified statement is anegative existential statement.

2 The negation of a existentially quantified statement is anegative universal statement.

For each sentence, write a quantified statement and its negation:

1 All cows have spots.

2 Some babies are born prematurely.

3 Pianos have 88 keys.

4 A bear is in the tree!




Summary


Practice: Negating Quantified Statements

1 All cows have spots.Statement: ∀x ∈ cows,Spotted(x)Negation: ∃ x ∈ cows, ∼ Spotted(x)

2 Some babies are born prematurely.Statement: ∃ x ∈ babies,Premature(x)Negation: ∀x ∈ babies, ∼ Premature(x)

3 Pianos have 88 keys.Statement: ∀x ∈ Instruments,Piano(x)→ Keys88(x)Negation: ∃ x ∈ Instruments,Piano(x) ∧ ∼ Keys88(x)

4 A bear is in the tree!Statement: ∃ x ∈ animals,Bear(x) ∧ inTree(x)Negation: ∀x ∈ animals,∼ Bear(x)∨ ∼ inTree(x)




Summary


Practice: Say it in English


1 ∀x ∈ squares,Rectangle(x).

2 ∃ y ∈ triangles, Isoceles(y)

3 ∀z ∈ USPresidents,Over35(z).

4 ∀x , y ∈ Z ,NonZero(x) ∧ NonZero(y)→ NonZero(x ∗ y).




Summary


Variations on a Universal Theme

Let B be the set of all birds.Let W(x) = x has wings.Let F(x) = x can fly.

Proposition Converse∀x ∈ B,W (x)→ F (x) ∀x ∈ B,F (x)→W (x)If a bird has wings, If a bird can fly,

then it can fly. then it has wings.

Inverse Contrapositive∀x ∈ B,∼W (x)→∼ F (x) ∀x ∈ B,∼ F (x)→ ∼W (x)If a bird does not have wings, If a bird cannot fly,

then it cannot fly. then it does not have wings.

Two of these are true, two are false. Which ones are which?




Summary


One or the Other is True.

A statement is true ↔ its negation is false.

Proof: Let p be any proposition.

p ∨ ∼ p Negation law.Assume ∼ p is false.Then p is true. Elimination

Now assume p is false.Then ∼ p is true. Elimination

∴ a statement is true iff its negation is false. Definition of ↔




Summary


Proving an Existential Statement

Symbolize this statement:

It is possible to get an “A” in this course.


Let A be the predicate x got an A in this course.

∃x ∈ P,A(x)

An existentially quantified predicate can be proved by finding asingle example that makes the statement true.

Sanjay is a person. Sanjay got an A in this course.Therefore, the statement is true.




Summary


(Dis)proving a Universal Statement

Symbolize this statement:

All people are good.


Let G be the predicate x is good.

∀x ∈ P,G (x)

To prove a universally quantified predicate you must show it is truefor all possible elements. It is often easier to disprove by finding asingle counterexample.

Hitler was a person. Hitler was not good.Therefore, the statement is not true.




Summary


Vacuous Truth

A universal statement can be true vacuously.

All purple cows with green spots eat scrap metal.

Unicorns are white with cream-colored horns.

A statement is true iff its negation is false. The negations are:

∃ a purple cow with green spots that does not eat scrap metal.

∃ a unicorn that is not white ∨ does not have a cream-coloredhorn.

These negations are false because unicorns and purple cows do notexist, ∴ the original statements are true.




Summary

Summary




Summary

Summary-1

Names for Sets of Numbers:

1 R, R−, R+: Real numbers, negative reals, positive reals.

2 Z , Z−, Z +: Integers, negative integers, positive integers.

3 Q, Q−, Q+: Rationals, negative rationals, positive rationals.

Note: Zero is not considered to be EITHER negative or positive.

Terminology:

The truth set of a predicate is the set of all values from therelevant domain that make the predicate true.

The propositional calculus deals with propositions (statementswith symbols, no variables no quantifiers).

The predicate calculus deals with predicates and quantifiersover sets of values.




Summary

Summary-2

Quantifiers:

1 ∀ is the universal quantifier and is read “for all”.

2 You can disprove a universally quantified statement by findingone counter-example.

3 ∃ is the existential quantifier and is read “there exists”.

4 You can disprove an existentially quantified statement byfinding one true example.




Summary

Quiz 5: Predicates

1. What is the truth set of this predicate if its domain is theintegers? x ∗ 2 < 10

2. In one word, what is the big difference between thepropositional calculus and the predicate calculus?

3. Symbolize the statement below.

Elderly (over 60) people are poor drivers.

4. Write the negative of the symbolic statement you created inproblem 3.

5. How you would go about proving or disproving it? (Justexplain how, you don’t actually have to do it)


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