CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables:...
Transcript of CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables:...
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CSE 20: Discrete Mathematics
Daniele Micciancio
Spring 2018
Daniele Micciancio CSE 20: Discrete Mathematics
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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.
Daniele Micciancio CSE 20: Discrete Mathematics
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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.
Daniele Micciancio CSE 20: Discrete Mathematics
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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.
Daniele Micciancio CSE 20: Discrete Mathematics
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Next question
Is the following deduction logically correct?
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.
Daniele Micciancio CSE 20: Discrete Mathematics
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Next question
Is the following deduction logically correct?
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.
Daniele Micciancio CSE 20: Discrete Mathematics
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Next question
Is the following deduction logically correct?
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.
Daniele Micciancio CSE 20: Discrete Mathematics
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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.
Daniele Micciancio CSE 20: Discrete Mathematics
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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.
Daniele Micciancio CSE 20: Discrete Mathematics
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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
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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
Daniele Micciancio CSE 20: Discrete Mathematics
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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
Daniele Micciancio CSE 20: Discrete Mathematics
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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
P(90, CSE20, CSE71)
(A) True; (B) False; (C) Undefined (Not a proposition)
Daniele Micciancio CSE 20: Discrete Mathematics
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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(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
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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(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
Daniele Micciancio CSE 20: Discrete Mathematics
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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
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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
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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
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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
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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
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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
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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?
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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?
Daniele Micciancio CSE 20: Discrete Mathematics
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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?
Daniele Micciancio CSE 20: Discrete Mathematics
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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)
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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 .
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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
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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
Note:
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
Daniele Micciancio CSE 20: Discrete Mathematics
![Page 29: CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables: P(x)=“x < 100” Q(x,y,z)=“x +y = z” R(x,y)=“Therearex studentsenrolledinclassy](https://reader035.fdocuments.us/reader035/viewer/2022071015/5fcdb78b41799933e73cb7cf/html5/thumbnails/29.jpg)
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
Note:
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
Daniele Micciancio CSE 20: Discrete Mathematics
![Page 30: CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables: P(x)=“x < 100” Q(x,y,z)=“x +y = z” R(x,y)=“Therearex studentsenrolledinclassy](https://reader035.fdocuments.us/reader035/viewer/2022071015/5fcdb78b41799933e73cb7cf/html5/thumbnails/30.jpg)
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
Note:
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
Daniele Micciancio CSE 20: Discrete Mathematics
![Page 31: CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables: P(x)=“x < 100” Q(x,y,z)=“x +y = z” R(x,y)=“Therearex studentsenrolledinclassy](https://reader035.fdocuments.us/reader035/viewer/2022071015/5fcdb78b41799933e73cb7cf/html5/thumbnails/31.jpg)
Order of quantification
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 .
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 .
Daniele Micciancio CSE 20: Discrete Mathematics
![Page 32: CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables: P(x)=“x < 100” Q(x,y,z)=“x +y = z” R(x,y)=“Therearex studentsenrolledinclassy](https://reader035.fdocuments.us/reader035/viewer/2022071015/5fcdb78b41799933e73cb7cf/html5/thumbnails/32.jpg)
Order of quantification
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 .
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 .
Daniele Micciancio CSE 20: Discrete Mathematics
![Page 33: CSE 20: Discrete Mathematics · Predicates Apredicate isalogicalstatementinvolvingvariables: P(x)=“x < 100” Q(x,y,z)=“x +y = z” R(x,y)=“Therearex studentsenrolledinclassy](https://reader035.fdocuments.us/reader035/viewer/2022071015/5fcdb78b41799933e73cb7cf/html5/thumbnails/33.jpg)
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)))
Daniele Micciancio CSE 20: Discrete Mathematics