cse 20: discrete mathematics · predicates apredicate isalogicalstatementinvolvingvariables:...

CSE 20: Discrete Mathematics

Daniele Micciancio

Spring 2018

Daniele Micciancio CSE 20: Discrete Mathematics

Summary

So far:

Propositional Logic: and, or, not, impliesReasoning: Truth tables, Equivalences, Proofs.

Today:

Predicate LogicExtend language with “every”, “some”, etc.Translating between English and Formal logicCarrying out proofsReading: Chap. 1.4, 1.5.

A classic example

Is the following deduction logically correct?

All men are mortalSocrates is a manTherefore, Socrates is mortal

(A) Yes; (B) No; (C) Not enough information; (D) I don’t know

The answer is (A), but we still do not have the tools to formallyjustify it.

A classic example

All men are mortalSocrates is a manTherefore, Socrates is mortal

(A) Yes; (B) No; (C) Not enough information; (D) I don’t know

The answer is (A), but we still do not have the tools to formallyjustify it.

Next question

All men are mortalSocrates is mortalTherefore, all man are Socrates

What about

All professors are grey at nightI am grey at nightTherefore I am a professor

Notes:

No propositional logic connectives. Atomic propositions.

To study this type of deductions we need to extend thelanguage of propositional logic.

Next question

What about

Notes:

Next question

What about

Notes:

Predicates

A predicate is a logical statement involving variables:

P(x) = “x < 100”Q(x,y,z) = “x + y = z”R(x,y) = “There are x students enrolled in class y”

Replacing the variable with concrete value yields propositions, whichmay be true or false:

P(33) = “33<100”Q(1,2,4) = “1 + 2 = 4”R(120,CSE12) = “There are 120 students enrolled in CSE12”

Predicates may be called “unary”, “binary”, n-ary, etc. dependingon how many variables they have.

Predicates

A predicate is a logical statement involving variables:

P(x) = “x < 100”Q(x,y,z) = “x + y = z”R(x,y) = “There are x students enrolled in class y”

Replacing the variable with concrete value yields propositions, whichmay be true or false:

P(33) = “33<100”Q(1,2,4) = “1 + 2 = 4”R(120,CSE12) = “There are 120 students enrolled in CSE12”

Predicates may be called “unary”, “binary”, n-ary, etc. dependingon how many variables they have.

Evaluating Predicates

Course Enrollment

CSE10 122CSE20 78CSE71 47

Q(x , y) = “There are at least x students enrolled in class y”R(y , z) = “Class y has smaller enrollment than class zP(x , y , z) = Q(x , y) ∧ R(z , y)

What is the truth value of

Q(90, CSE20)

(A) True; (B) False; (C) Undefined (Not a proposition)

B: There are less than 90 students in CSE20

Course Enrollment

CSE10 122CSE20 78CSE71 47

Q(90, CSE20)

Course Enrollment

CSE10 122CSE20 78CSE71 47

Q(90, CSE20)

Course Enrollment

CSE10 122CSE20 78CSE71 47

P(90, CSE20, CSE71)

Course Enrollment

CSE10 122CSE20 78CSE71 47

Q(x , y) = “There are at least x students enrolled in class y”R(y , z) = “Class y has smaller enrollment than class zP(x , y , z) = Q(x , y) ∧ R(y , z)

For what values of x , y , z , is the following predicate true?

P(x , y , z)→ Q(x , z)

(A) Always true; (B) Never true; (C) Depends on x , y , z

A: class z is larger than class y, and class y has at least x students

Course Enrollment

CSE10 122CSE20 78CSE71 47

Q(x , y) = “There are at least x students enrolled in class y”R(y , z) = “Class y has smaller enrollment than class zP(x , y , z) = Q(x , y) ∧ R(y , z)

For what values of x , y , z , is the following predicate true?

P(x , y , z)→ Q(x , z)

(A) Always true; (B) Never true; (C) Depends on x , y , z

A: class z is larger than class y, and class y has at least x students

Universal quantifier (for all)

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

∀x , P(x) = “For all (integers) x , P(x) is true.”

∀x , Q(x) = “For all (integers) x , Q(x) is true.”

Think of ∀ as an upside-down “A” (for “All”)May also be written ∀x .P(x), or ∀xP(x)∀x .P(x) and ∀x .Q(x) are logical statements (either true orfalse)∀x .P(x) is equivalent to ∀y .P(y). (The variable name doesnot matter.)

Are ∀x .P(x) and ∀x .Q(x) true or false?

(A) T T ; (B) T F ; (C) F T ; (D) F F

B: True and False

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

(A) T T ; (B) T F ; (C) F T ; (D) F F

B: True and False

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

(A) T T ; (B) T F ; (C) F T ; (D) F F

B: True and False

Existential quantifier (there exists)

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

∃x , P(x) = “For some (integers) x , P(x) is true.”

∃x , Q(x) = “For some (integers) x , Q(x) is true.”

Think of ∃ as an “E” (for “Exists”) written backwardMay also be written ∃x .P(x), or ∃xP(x)∃x .P(x) and ∃x .Q(x) are logical statements (either true orfalse)∃x .P(x) is equivalent to ∃y .P(y). (The variable name doesnot matter.)

Are ∃x .P(x) and ∃x .Q(x) true or false?

(A) T T ; (B) T F ; (C) F T ; (D) F F

A: True and True

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

(A) T T ; (B) T F ; (C) F T ; (D) F F

A: True and True

P(x) = (x > 5)→ (x2 > 25)Q(x) = (x < 5)→ (x2 < 25)

(A) T T ; (B) T F ; (C) F T ; (D) F F

A: True and True

Tautologies in Predicate calculus

(∀x .(Q(x) ∧ P(x)))→ (∀y .Q(y))(∃x .(P(x)→ Q(x))) ∧ (∀x .P(x))→ ∃x .Q(x)(∃x .(P(x) ∨ Q(x)))↔ (∃x .P(x)) ∨ (∃x .Q(x))

For what direction the following implication holds?

(∀x .(P(x) ∨ Q(x))) [↔???] (∀x .P(x)) ∨ (∀x .Q(x))

(A) ↔ (B) Only →; (C) Only ← ; (D) Neither direction

Answer: C

Why? Let x be an integer values variable, and let

P(x)=“x is an even integer”Q(x)=“x is an odd integer”

How would you read the quantified statements in English?

(∀x .(P(x) ∨ Q(x))) [↔???] (∀x .P(x)) ∨ (∀x .Q(x))

Answer: C

(∀x .(P(x) ∨ Q(x))) [↔???] (∀x .P(x)) ∨ (∀x .Q(x))

Answer: C

Quantifying over Finite Domains

Assume variable x ranges over a finite set {1, 2, 3}.

∀x .P(x) ⇐⇒ P(1) ∧ P(2) ∧ P(3)

∃x .P(x) ⇐⇒ P(1) ∨ P(2) ∨ P(3)

Most logic rules about ∀ and ∃ can be understood in terms of ∧and ∨.

De Morgan:

¬(∀x .P(x))

⇐⇒ ¬(P(1) ∧ P(2) ∧ P(3))

⇐⇒ (¬P(1) ∨ ¬P(2) ∨ ¬P(3))

⇐⇒ ∃x .¬P(x)

Negating quantifiers

¬∀x .P(x) ≡ ∃x .¬P(x)

It is not true that P(x) holds for every xThere is some x for which P(x) is not true

¬∃x .P(x) ≡ ∀x .¬P(x)

It is not true that P(x) holds for some xP(x) is false for every x .

Proving and disproving quantified statements

∀x .P(x)

Prove: Need to show P(x) for an arbitrary xDisprove: Enough to show that ¬P(x) for some specific x ofour choice

∃x .P(x)

Prove: Enough to show P(x) is true for some specific x of ourchoiceDisprove: Need to show that P(x) is false for an arbitrary x

Nested quantifiers

Let’s talk about integer numbers. (Variables x , y , z range over Z).

“For every integer x , there is always some bigger integer y .

∀x .∃y .y > x

Q(x , y) = (y > x) is a binary predicateP(x) = ∃y .Q(x , y) is a unary predicate∀x .P(x) ≡ ∀x .∃y .Q(x , y) is a proposition

Is the statement ∀x .∃y .Q(x , y) true or false?

(A) True; (B) False; (C) It depends on the value of x

Nested quantifiers

∀x .∃y .y > x

Nested quantifiers

∀x .∃y .y > x

Order of quantification

∀x .∃y .y > x .

What happen if we swap the order of quantifiers?

∃y .∀x .y > x .

Are the two logical statements equivalent?Is the new statement true or false?How would you read the last statement in English?

“There is an integer y which is bigger than any (other) integer x .

Order of quantification

∀x .∃y .y > x .

What happen if we swap the order of quantifiers?

∃y .∀x .y > x .

Are the two logical statements equivalent?Is the new statement true or false?How would you read the last statement in English?

“There is an integer y which is bigger than any (other) integer x .

Scope and Precedence

Textbook: ∀,∃ have the highest “precedence”:

∃x .(∀x .P(x) ∧ Q(x))

⇐⇒ ∃x .((∀x .P(x)) ∧ Q(x))

⇐⇒ (∀x .P(x)) ∧ (∃x .Q(x))

This is unusual. Most texts let ∀x and ∃x extend their scope as faras possible, unless limited with parentheses.

∃x .(∀x .P(x) ∧ Q(x))

⇐⇒ ∃x .(∀x .(P(x) ∧ Q(x)))

⇐⇒ (∀x .(P(x) ∧ Q(x)))

cse 20: discrete mathematics · predicates apredicate isalogicalstatementinvolvingvariables:...

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