[Hernandez 1213A] Cs30 day1

Propositional Calculus

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 1

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Mathematical Logic

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1

Propositional Calculus Introduction Statements and Connectives

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."

Lewis Carroll, Through the Looking-Glass, Chapter 4

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

Logic is the technique by which we add conviction to truth.

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Jean De La Bruyere

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

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Logic is logic. That's all I say.

Oliver Wendell Holmes, The One-Hoss Shay

http://www.ibiblio.org/eldritch/owh/shay.html

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

All men are mortal. Socrates was mortal. Therefore, all men are Socrates. Woody Allen

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

All men are mortal. Socrates was mortal. Therefore, all men are Socrates. Woody Allen To love is to su er. To avoid su ering one must not love, but then one su ers from not loving. Therefore, to love is to su er, not to love is to su er, to su er is to su er. To be happy is to love, to be happy then is to su er but su ering makes one unhappy, therefore to be unhappy one must love or love to su er or su er from too much happiness. I hope you're getting this down... Diane Keaton

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Love and Death updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

What is LOGIC ???

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Grammar is the logic of speech, even as logic is the grammar of reason.

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Richard Chenevix Trench

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

A language in which we describe another language is called a metalanguage.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

A language in which we describe another language is called a metalanguage. For almost all of mathematics, the metalanguage is English with some extra notation.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

A language in which we describe another language is called a metalanguage. For almost all of mathematics, the metalanguage is English with some extra notation.

In computing we need to de ne, and use, languages and formal notation so it is essential that we have a clear and precise metalanguage.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

A language in which we describe another language is called a metalanguage. For almost all of mathematics, the metalanguage is English with some extra notation.

In computing we need to de ne, and use, languages and formal notation so it is essential that we have a clear and precise metalanguage. We begin

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by looking at some English expressions which we could use in computing.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Language? Metalanguage?

A language in which we describe another language is called a metalanguage. For almost all of mathematics, the metalanguage is English with some extra notation.

In computing we need to de ne, and use, languages and formal notation so it is essential that we have a clear and precise metalanguage. We begin

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by looking at some English expressions which we could use in computing.

If it is raining and I go outside then I get wet.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition)

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2?

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

3

This sentence is false.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

3

This sentence is false. NO!

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

3

This sentence is false. NO!

4

Today is Monday.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

3

This sentence is false. NO!

4

Today is Monday. YES!

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updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Statement (also, Proposition) A proposition (or, statement) is a declarative sentence that is either true or false, but not both.

Example 1

What is 2+2? NO!

2

Evaluate the sum 2+2. NO!

3

This sentence is false. NO!

4

Today is Monday. YES!

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We use letters to denote propositional variables (or statement variables).

The truth value of a proposition is true, denoted by T, if it is a true proposition; and false, denoted by F, if it is a false proposition.

The area of logic that deals with propositions is called the propositional calculus or propositional logic.

Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Many mathematical statements are constructed by combining one or

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more propositions.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Many mathematical statements are constructed by combining one or

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more propositions.

New propositions, called compound propositions, are formed from existing propositions using logical operators (or, connectives).

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Negation,

ÂŹ

or

âˆź

The negation of a statement is false when the statement is true and is true if the statement is false.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Negation,

ÂŹ

or

âˆź

The negation of a statement is false when the statement is true and is true if the statement is false.

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Example Statement: All computer scientists are men.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Negation,

ÂŹ

or

âˆź

The negation of a statement is false when the statement is true and is true if the statement is false.

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Example Statement: All computer scientists are men.

Negation of statement?

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Negation,

ÂŹ

or

âˆź

The negation of a statement is false when the statement is true and is true if the statement is false.

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Example Statement: All computer scientists are men.

Negation of statement?

No computer scientists are men.

Some computer scientists are men.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Negation,

¬

or

∼

The negation of a statement is false when the statement is true and is true if the statement is false.

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Example Statement: All computer scientists are men.

Negation of statement?

No computer scientists are men.

Some computer scientists are men.

p ∼p T F F T Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

and

q

∧

are statements, then

p∧q

is read as

p

and

q

.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

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p: It is green. q: It is an apple.

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

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p: It is green.

p∧q

q: It is an apple. :

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

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p: It is green.

p∧q

q: It is an apple. : It is green and it is an apple.

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

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p: It is green.

p∧q

q: It is an apple. : It is green and it is an apple.

p q p∧q T T T

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

sablay-logo

p: It is green.

p∧q

q: It is an apple. : It is green and it is an apple.

p q p∧q T T T T F F

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

sablay-logo

p: It is green.

p∧q

q: It is an apple. : It is green and it is an apple.

p T T F

q p∧q T T F F T F

Discrete Mathematics for Computer Science

CS 30

updilseal

dcs-logo

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conjunction, If

p

q

and

∧

are statements, then

p∧q

is read as

Example

p

and

q

.

sablay-logo

p: It is green.

p∧q

q: It is an apple. : It is green and it is an apple.

p T T F F

q p∧q T T F F T F F F

Discrete Mathematics for Computer Science

CS 30

updilseal

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

and

q

∨

are statements, then

p∨q

is read as

p q or

.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

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p: I am a teacher. q: I am a student.

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. :

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. : I am a teacher or I am a student.

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. : I am a teacher or I am a student.

p q p∨q T T T

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. : I am a teacher or I am a student.

p q p∨q T T T T F T

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. : I am a teacher or I am a student.

p T T F

q p∨q T T F T T T

Discrete Mathematics for Computer Science

CS 30

updilseal

dcs-logo

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Disjunction, If

p

q

and

∨

are statements, then

p∨q

is read as

Example

p q or

.

sablay-logo

p: I am a teacher.

p∨q

q: I am a student. : I am a teacher or I am a student.

p T T F F

q p∨q T T F T T T F F

Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Exclusive OR, Let also

p q p 6≡ q and

⊕

or

6≡

be statements. The

exclusive p of

and

q

true and is false otherwise.

p

, denoted by

, is the proposition that is true when exactly one of

p⊕q q

and

is

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Exclusive OR, Let also

p q p 6≡ q and

⊕

or

6≡

be statements. The

exclusive p of

and

q

true and is false otherwise.

p T T F F

q p 6≡ q T F F T T T F F

p

, denoted by

, is the proposition that is true when exactly one of

p⊕q q

and

is

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

if

p p

and

then

q q

.

conditional statement p =⇒ q

is read as

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

if

p p

and

then

q q p .

is called the

conditional statement p =⇒ q hypothesis antecendent premise

is read as

(or

or

)

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

q

if

p p

and

then

q conditional statement p =⇒ q q p hypothesis antecendent premise conclusion consequence .

is called the

is called the

(or

is read as

(or

or

) and

).

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updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

q

if

p p

and

then

q conditional statement p =⇒ q q p hypothesis antecendent premise conclusion consequence .

is called the

is called the

(or

is read as

(or

or

) and

).

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Example p: You got a nal grade of 93. q: You got a 1.0.

updilseal

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Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

q

if

p p

and

then

q conditional statement p =⇒ q q p hypothesis antecendent premise conclusion consequence .

is called the

is called the

(or

is read as

(or

or

) and

).

sablay-logo

Example p: You got a nal grade of 93.

p =⇒ q

q: You got a 1.0. :

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

q

if

p p

and

then

q conditional statement p =⇒ q q p hypothesis antecendent premise conclusion consequence .

is called the

is called the

(or

is read as

(or

or

) and

).

sablay-logo

Example p: You got a nal grade of 93.

p =⇒ q

q: You got a 1.0. : If you got a nal grade of 93, then you got a 1.0.

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Conditional,

=⇒

Let

be statements. The

q

if

p p

and

then

q conditional statement p =⇒ q q p hypothesis antecendent premise conclusion consequence .

is read as

is called the

is called the

(or

(or

or

) and

).

sablay-logo

Example p: You got a nal grade of 93.

p =⇒ q

q: You got a 1.0. : If you got a nal grade of 93, then you got a 1.0.

p T T F F

q p =⇒ q T T F F T T F T

Discrete Mathematics for Computer Science

CS 30

updilseal

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Propositional Calculus

Introduction Statements and Connectives

Terminology : if p, then q p implies q q if p p only if q q when p p is su cient for q a necessary condition for p is q a su cient condition for q is p q whenever p q is necessary for p q follows from p q unless ÂŹp

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Consider these ...

If the sun shines today we will have a barbecue.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Consider these ...

If the sun shines today we will have a barbecue.

If the sun shines today 19 is prime.

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Consider these ...

If the sun shines today we will have a barbecue.

If the sun shines today 19 is prime.

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If the sun shines today 8 is prime.

updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

converse

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

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updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

contrapositive

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

contrapositive

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¬ =⇒ ¬

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CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

contrapositive

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¬ =⇒ inverse

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

p

q

contrapositive

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¬ =⇒ inverse

¬ =⇒ ¬

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

p

q

contrapositive

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¬ =⇒ inverse

¬ =⇒ ¬

When two compound propositions always have the same truth value we call them equivalent. updilseal

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

p

q

contrapositive

sablay-logo

¬ =⇒ inverse

¬ =⇒ ¬

When two compound propositions always have the same truth value we call them equivalent.

That is, a conditional statement and its contrapositive are equivalent.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Forming new conditional statements starting with a conditional statement

Introduction Statements and Connectives

p =⇒ q

q =⇒ p

converse

q

p

p

q

contrapositive

sablay-logo

¬ =⇒ inverse

¬ =⇒ ¬

When two compound propositions always have the same truth value we call them equivalent.

That is, a conditional statement and its contrapositive are equivalent.

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Also, the converse and inverse of a conditional statement are equivalent. dcs-logo

Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Biconditional, Let as

p p

and

q

�⇒

su cient for

q

p q

be statements. The

q

if and only if .

or

i

biconditional statement p �⇒ q p

. It is also read as

is read

is necessary and

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Biconditional, Let as

p p

and

q

�⇒

su cient for

q

p q

be statements. The

q

if and only if .

or

i

biconditional statement p �⇒ q p

. It is also read as

is read

is necessary and

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Example p: You are enrolled in CS 30. q: You have passed Math 17.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Biconditional, Let as

p p

and

q

�⇒

su cient for

q

p q

be statements. The

q

if and only if .

or

i

biconditional statement p �⇒ q p

. It is also read as

is read

is necessary and

sablay-logo

Example p: You are enrolled in CS 30.

p �⇒ q

q: You have passed Math 17. :

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Biconditional, Let as

p p

and

q

�⇒

su cient for

q

p q

be statements. The

q

if and only if .

or

i

biconditional statement p �⇒ q p

. It is also read as

is read

is necessary and

sablay-logo

Example p: You are enrolled in CS 30.

p �⇒ q

q: You have passed Math 17. : You are enrolled in CS 30 if and only if you have passed

Math17.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Connectives ? Symbolic Notations?

Biconditional, Let as

p p

and

q

�⇒

su cient for

q

p q

be statements. The

q

if and only if .

or

i

biconditional statement p �⇒ q p

. It is also read as

is read

is necessary and

sablay-logo

Example p: You are enrolled in CS 30.

p �⇒ q

q: You have passed Math 17. : You are enrolled in CS 30 if and only if you have passed

Math17.

p T T F F

q p �⇒ q T T F F T F F T

Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Precedence of Logical Operators In logical expressions involving more than one operation, the order of operations must be understood.

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CS 30

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Propositional Calculus

Introduction Statements and Connectives

Precedence of Logical Operators In logical expressions involving more than one operation, the order of operations must be understood.

The order of precedence of the basic operations listed from highest to lowest is

¬

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∧ ∨ =⇒ ⇐⇒

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Precedence of Logical Operators In logical expressions involving more than one operation, the order of operations must be understood.

The order of precedence of the basic operations listed from highest to lowest is

¬

sablay-logo

∧ ∨ =⇒ ⇐⇒ Note: In programming languages such as C/C++, as

p q r

( ∧ )∨

, where

∧

is denoted

&&

and

p&&q||r ∨

would be interpreted

is denoted

||.

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

Precedence of Logical Operators In logical expressions involving more than one operation, the order of operations must be understood.

The order of precedence of the basic operations listed from highest to lowest is

¬

sablay-logo

∧ ∨ =⇒ ⇐⇒ Note: In programming languages such as C/C++, as

p q r

( ∧ )∨

, where

∧

is denoted

p∧q∨r

&&

and

p&&q||r ∨

would be interpreted

is denoted

||.

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However, in applications such as Microsoft Excel, it is impossible to enter

p q r p ∧ (q ∨ r )

an expression of the form

( ∧ )∨

or

.

Discrete Mathematics for Computer Science

without explicitly entering it either as

CS 30

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Propositional Calculus

Truth Table of

p∧q∨r

Introduction Statements and Connectives

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CS 30

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Propositional Calculus

p∧q∨r p q r p ∧ q (p ∧ q ) ∨ r T T T T F F F F

Introduction Statements and Connectives

Truth Table of

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CS 30

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Propositional Calculus

p∧q∨r q r p ∧ q (p ∧ q) ∨ r T T F F T T F F

Introduction Statements and Connectives

Truth Table of

p T T T T F F F F

sablay-logo

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

p∧q∨r r p ∧ q (p ∧ q) ∨ r T F T F T F T F

Introduction Statements and Connectives

Truth Table of

p T T T T F F F F

q T T F F T T F F

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

p∧q∨r r p ∧ q (p ∧ q) ∨ r T T F T T F F F T F F F T F F F

Introduction Statements and Connectives

Truth Table of

p T T T T F F F F

q T T F F T T F F

sablay-logo

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

p∧q∨r r p ∧ q (p ∧ q) ∨ r T T T F T T T F T F F F T F T F F F T F T F F F

Introduction Statements and Connectives

Truth Table of

p T T T T F F F F

q T T F F T T F F

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

Introduction Statements and Connectives

A Logic Puzzle

Knights and Knaves The logician Raymond Smullyan describes an island containing two types

knights who always tell the truth and knaves who always lie.

of people:

You visit the island and are approached by two natives who speak to you as follows: A says: B says:

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B is a knight.

A and I are of opposite type.

What are A and B ?

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Discrete Mathematics for Computer Science

CS 30

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Propositional Calculus

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Questions?

See you next meeting!

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Discrete Mathematics for Computer Science

CS 30

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