Propositional Calculus (cont...)
CS 30 : Discrete Mathematics for Computer Science
First Semester, AY 2012-2013
https://sites.google.com/a/dcs.upd.edu.ph/nhsh_classes/cs 30
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Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph
updilseal Day 2
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Mathematical Logic
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1
Propositional Calculus (cont...) Arguments Predicates and Quanti ers
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction?
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology.
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updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
âˆźp
is
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
âˆźp
is a tautology.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
∼p
is a tautology.
2
The statement p ∧
∼p
is
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
∼p
is a tautology.
2
The statement p ∧
∼p
is a contradiction.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
sablay-logo
Example 1
The statement p ∨
∼p
is a tautology.
2
The statement p ∧
∼p
is a contradiction.
3
The statement
((p ∨ q ) ∧ p ) ⇐⇒ p
is
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
sablay-logo
Example 1
The statement p ∨
∼p
is a tautology.
2
The statement p ∧
∼p
is a contradiction.
3
The statement
((p ∨ q ) ∧ p ) ⇐⇒ p
is a tautology.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
âˆźp
is a tautology.
2
The statement p ∧
âˆźp
is a contradiction.
3
The statement
((p ∨ q ) ∧ p ) �⇒ p
is a tautology.
Remark: p1 �⇒ p2 is a tautology i p1 and p2 have the same truth value. updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Tautology? Contradiction? De nition A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
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Example 1
The statement p ∨
âˆźp
is a tautology.
2
The statement p ∧
âˆźp
is a contradiction.
3
The statement
((p ∨ q ) ∧ p ) �⇒ p
is a tautology.
Remark: p1 �⇒ p2 is a tautology i p1 and p2 have the same truth value. We say that p1 and p2 are equivalent, denoted p1 ≥ p2 .
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument ?
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument ? De nition A conditional of the form (a conjunction of statements) implies c where c is a statement, is called an argument . That is,
p1
∧ p2 ∧ . . . ∧ pn =⇒ c
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The statements in the conjunction on the left side of the conditional are called premises , while c is called the conclusion.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid?
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid? An argument is valid if it is a tautology.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid? An argument is valid if it is a tautology.
p1
∧ p2 ∧ . . . ∧ pn =⇒ c
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid? An argument is valid if it is a tautology.
p1
∧ p2 ∧ . . . ∧ pn =⇒ c
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Note: 1
A conjunction of several statements is true only when all the statements are true.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid? An argument is valid if it is a tautology.
p1
∧ p2 ∧ . . . ∧ pn =⇒ c
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Note: 1
A conjunction of several statements is true only when all the statements are true.
2
A conditional is false only when the antecedent is true and the consequence is false.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Argument valid? An argument is valid if it is a tautology.
p1
∧ p2 ∧ . . . ∧ pn =⇒ c
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Note: 1
A conjunction of several statements is true only when all the statements are true.
2
A conditional is false only when the antecedent is true and the consequence is false.
3
Therefore, an argument is invalid only when there is an instance where all the premises are true, but the conclusion is false. If such a situation cannot occur, the argument is valid.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables 1
Translate the premises and the conclusion into symbolic form.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables 1
Translate the premises and the conclusion into symbolic form.
2
Write the truth table for the premises and the conclusion.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables 1
Translate the premises and the conclusion into symbolic form.
2
Write the truth table for the premises and the conclusion.
3
Determine if there is a row in which all the premises are true and the conclusion is false. If yes, the argument is invalid, otherwise it is
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valid.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables 1
Translate the premises and the conclusion into symbolic form.
2
Write the truth table for the premises and the conclusion.
3
Determine if there is a row in which all the premises are true and the conclusion is false. If yes, the argument is invalid, otherwise it is
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valid.
Example 1
, , ((p ∧ q ) ⇒∼ r ) =⇒∼ q
p r
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Truth Tables 1
Translate the premises and the conclusion into symbolic form.
2
Write the truth table for the premises and the conclusion.
3
Determine if there is a row in which all the premises are true and the conclusion is false. If yes, the argument is invalid, otherwise it is
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valid.
Example 1
p r
2
Mary studies hard. If Mary studies hard then she is a nerd. If Mary
, , ((p ∧ q ) ⇒∼ r ) =⇒∼ q
is a nerd then she does not party. Therefore Mary does not party. updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
A more complex deduction You are about to leave for school in the morning and discover that you don't have your glasses. You know the following statements are true: a. If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table. b. If my glasses are on the kitchen table, then I saw them at
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breakfast. c. I did not see my glasses at breakfast. d. I was reading the newspaper in the living room or I was reading the newspaper in the kitchen. e. If I was reading the newspaper in the living room then my glasses are on the co ee table. Now I know where my glasses are! It's on the co ee table! :-)
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Analyzing Arguments Using Rules of Inference RULES OF INFERENCE
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p, q , r , s
can be any statement.
c
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is a contradiction.
Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Try this ... If I go to my rst class tomorrow, then I must get up early. If I go to the dance tonight, I will stay up late. If I stay up late and get up early, then I will be forced to exist on only ve hours sleep. I cannot exist on ve hours of sleep. Therefore I must either miss my rst class tomorrow or not go to the
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dance.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Logical Equivalences
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t
is a tautology.
c
is a contradiction.
p, q , r
can be any statement.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Logical Equivalences
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t
is a tautology.
c
is a contradiction.
p, q , r
can be any statement.
Knowledge of logically equivalent statements is very useful for constructing arguments. It often happens that it is di cult to see how a conclusion follows from one form of a statement, whereas it is easy to see how it follows from a logically equivalent form of the statement. Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
More Logical Equivalences 1
p
2
p
→ q ≡ ¬p ∨ q
3
p
∧ q ≡ ¬(p → ¬q )
4
¬(p → q ) ≡ p ∧ ¬q
5
(p → q ) ∧ (p → r ) ≡ p → (q ∧ r )
6
(p → r ) ∧ (q → r ) ≡ (p ∨ q ) → r
7
(p → q ) ∨ (p → r ) ≡ p → (q ∨ r )
8
(p → r ) ∨ (q → r ) ≡ (p ∧ q ) → r
9
p
↔ q ≡ (p → q ) ∧ (q → p )
10
p
↔ q ≡ ¬p ↔ ¬q
11
p
↔ q ≡ (p ∧ q ) ∨ (¬p ∧ ¬q )
12
¬(p ↔ q ) ≡ p ↔ ¬q
→ q ≡ ¬q → ¬p
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Going back to ... Knights and Knaves The logician Raymond Smullyan describes an island containing two types of people: knights who always tell the truth and knaves who always lie.
You visit the island and are approached by two natives who speak to you
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as follows: A says: B is a knight. B says: A and I are of opposite type. You say: Both of you are surely knaves!
Give an argument as to why you arrive at that conclusion.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Predicate ? De nition A
predicate is a statement containing one or more variables.
If values
are assigned to all the variables in a predicate, the resulting statement is a proposition.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Predicate ? De nition A
predicate is a statement containing one or more variables.
If values
are assigned to all the variables in a predicate, the resulting statement is a proposition.
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Example x
<5
is a predicate, where x is a variable denoting any real number.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Predicate ? De nition A
predicate is a statement containing one or more variables.
If values
are assigned to all the variables in a predicate, the resulting statement is a proposition.
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Example x
<5
is a predicate, where x is a variable denoting any real number.
Substituting a real number for x , yields a proposition:
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Predicate ? De nition A
predicate is a statement containing one or more variables.
If values
are assigned to all the variables in a predicate, the resulting statement is a proposition.
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Example x
<5
is a predicate, where x is a variable denoting any real number.
Substituting a real number for x , yields a proposition: 3
<5
6
<5 updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Predicate ? De nition A
predicate is a statement containing one or more variables.
If values
are assigned to all the variables in a predicate, the resulting statement is a proposition.
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Example x
<5
is a predicate, where x is a variable denoting any real number.
Substituting a real number for x , yields a proposition: 3
<5
6
<5 updilseal
What are their truth values?
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Quanti ers ? Quanti ers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Quanti ers ? Quanti ers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true.
â&#x2C6;&#x20AC;
universal quanti er
â&#x2C6;&#x192;
existential quanti er
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Quanti ers ? Quanti ers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true.
â&#x2C6;&#x20AC;
universal quanti er
â&#x2C6;&#x192;
existential quanti er
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Example There exists an x such that x
< 5.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Quanti ers ? Quanti ers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true.
∀
universal quanti er
∃
existential quanti er
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Example There exists an x such that x
< 5.
∃x , x < 5
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Quanti ers ? Quanti ers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true.
∀
universal quanti er
∃
existential quanti er
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Example There exists an x such that x
< 5.
∃x , x < 5 For all x , x
<5
or x
≥ 5.
∀x , (x < 5 ∨ x ≥ 5)
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Universal Statement? Existential Statement? De nition Let Q (x ) be a predicate and D the domain of x . A
universal statement is a statement of the form â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; D , Q (x ).
It is de ned to be true if and only if Q (x ) is true for every x in D .
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It is de ned to be false if and only if Q (x ) is false for at least one x in D .
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Universal Statement? Existential Statement? De nition Let Q (x ) be a predicate and D the domain of x . A
universal statement is a statement of the form â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; D , Q (x ).
It is de ned to be true if and only if Q (x ) is true for every x in D .
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It is de ned to be false if and only if Q (x ) is false for at least one x in D . A value for x for which Q (x ) is false is called a counterexample to the universal statement.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Universal Statement? Existential Statement? De nition Let Q (x ) be a predicate and D the domain of x . A
universal statement is a statement of the form ∀x ∈ D , Q (x ).
It is de ned to be true if and only if Q (x ) is true for every x in D .
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It is de ned to be false if and only if Q (x ) is false for at least one x in D . A value for x for which Q (x ) is false is called a counterexample to the universal statement. An
existential statement is a statement of the form ∃x ∈ D
such that
( )
Q x .
It is de ned to be true if and only if Q (x ) is true for at least one x in D. It is false if and only if Q (x ) is false for all x in D .
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2
Consider the statement
∀x ∈ R, x 2 ≥ x .
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2
Consider the statement
∀x ∈ R, x 2 ≥ x .
FALSE
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Discrete Mathematics for Computer Science
CS 30
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Arguments Predicates and Quanti ers
Propositional Calculus (cont...)
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2 3
Consider the statement Consider the statement
∀x ∈ R, x 2 ≥ x .
∃m ∈ Z
+
FALSE
such that m
2
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= m.
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Discrete Mathematics for Computer Science
CS 30
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Arguments Predicates and Quanti ers
Propositional Calculus (cont...)
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2 3
Consider the statement Consider the statement
∀x ∈ R, x 2 ≥ x .
∃m ∈ Z
+
FALSE
such that m
2
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= m.
TRUE
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Discrete Mathematics for Computer Science
CS 30
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Arguments Predicates and Quanti ers
Propositional Calculus (cont...)
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2 3 4
Consider the statement Consider the statement
= {5, 6, 7, 8} = m.
Let E m
2
∀x ∈ R, x 2 ≥ x .
∃m ∈ Z
+
FALSE
and consider the
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2
= m. TRUE statement ∃m ∈ E such
such that m
that
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Discrete Mathematics for Computer Science
CS 30
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Arguments Predicates and Quanti ers
Propositional Calculus (cont...)
Truth Value of a Quanti ed Statement Example 1
Let D
= {1, 2, 3, 4, 5}
and consider the statement
∀x ∈ D , x 2 ≥ x .
TRUE 2 3 4
Consider the statement Consider the statement
= {5, 6, 7, 8} = m. FALSE
Let E m
2
∀x ∈ R, x 2 ≥ x .
∃m ∈ Z
+
FALSE
and consider the
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2
= m. TRUE statement ∃m ∈ E such
such that m
that
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Validity of Arguments with Quanti ed Statements Using diagrams to show validity Example All human beings are mortal. Zeus is not mortal.
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Therefore, Zeus is not a human being.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Validity of Arguments with Quanti ed Statements Using diagrams to show invalidity Example All human beings are mortal. Felix is mortal.
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Therefore, Felix is a human being.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Validity of Arguments with Quanti ed Statements Using rules of inference and/or logical equivalences Example All the triangles are blue. If an object is to the right of all the squares, then it is above all the
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circles. If an object is not to the right of all the squares, then it is not blue. Therefore, all the triangles are above all the circles.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Validity of Arguments with Quanti ed Statements Using rules of inference and/or logical equivalences Let T (x ) be "x is a triangle";
( ) be "x is a square"; R (x , y ) be "x is to the right
( ) be "x is blue"; ( ) be "x is a circle"; A(x , y ) be "x is above
B x
S x
C x
of y";
y".
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All the triangles are blue.
∀x , {T (x ) ⇒ B (x )} If an object is to the right of all the squares, it is above all the circles.
∀x , {∀y (S (y ) ⇒ R (x , y )) ⇒ ∀z (C (z ) ⇒ A(x , z ))} If an object is not to the right of all the squares, then it is not blue.
∀x , {¬ (∀y (S (y ) ⇒ R (x , y ))) ⇒ ¬B (x )} Therefore, all the triangles are above all the circles.
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∀x , {T (x ) ⇒ ∀z (C (z ) ⇒ A(x , z ))} Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
Now try this ... Prove that the sum of two even integers is even.
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Arguments Predicates and Quanti ers
On ambiguity ...
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updilseal
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Discrete Mathematics for Computer Science
CS 30
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Propositional Calculus (cont...)
Questions?
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See you next meeting!
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Discrete Mathematics for Computer Science
CS 30
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