discreet structures ch1-1

8/10/2019 Discreet Structures Ch1-1

1/48

EECS 1590-002

Discrete Structures

Chapter 1, Section 1.1Propositional Logic

These class notes are based on material from our

textbook, Discrete Mathematics and ItsApplications, by Kenneth H. Rosen. They areintended for classroom use only and are notasubstitute for reading the textbook.


2/48

Proposition

Statement that is trueorfalse.

Examples:

The capital of New York is Albany.The moon is made of green cheese.

Go to town. (not a propositionwhy?)

What time is it? (not a propositionwhy?)

x+ 1 = 2 (not a propositionwhy?)


3/48

Simple/Compound Propositions

A simple proposition has a value of T/F

A compound proposition is constructed from

one or more simple propositions using logical

operators

The truth value of a compound proposition

depends on the truth values of the constituentpropositions


4/48

Negation (NOT)

NOTcan be represented by the ~ or

symbols

NOT is a logical operator:

p: I am going to town.

~p: I am not going to town.


5/48


6/48

Truth Tables

A truth table lists ALL possible values of a

(compound) proposition

one column for each propositional variable

one column for the compound proposition

2nrows for npropositional variables


7/48

Conjunction (AND)

The conjunctionAND is a logical operator


q: It is raining.

pq: I am going to town and it is raining.

Bothpand qmust be true for the conjunctionto be true.


8/48

Truth table for ^ (AND)

p q p q

T T T

T F F

F T F

F F F


9/48

Disjunction (OR)

Inclusive or - only one proposition needs to be

true for the disjunction to be true.


q: It is raining.

pq: I am going to town or it is raining.


10/48

Truth table for (OR)

p q p q

T T T

T F T

F T T

F F F


11/48

Exclusive OR

Only one ofpand qare true (not both).


q: It is raining.

pq: Either I am going to town or it is raining.


12/48

Truth table for (Exclusive OR)

p q p q

T T F

T F T

F T T

F F F


13/48

Conditional statements

A conditionalstatement is also called an

implicationor an if .. then statement.

It has the formp q


q: It is raining.

pq: If I am going to town, then it is raining.

The implication is false only whenpis true and

qis false!


14/48

Truth table for Conditional statements

p q p q

T T T

T F F

F T T

F F T


15/48

Implication - Equivalent Forms

Ifp, then q

pimplies q

Ifp, q

qifp

qwheneverp

pis a sufficient condition for q

qis a necessary condition forp

ponly if q


16/48

Converse of an Implication

Implication: pq

Converse: qp

Implication:

If I am going to town, it is raining.

Converse:

If it is raining, then I am going to town.


17/48

Converse of an Implication

p q q p

T T T

T F T

F T F

F F T

(for pq, this would be F)

(for pq, this would be true)


18/48

Contrapositive of an Implication

Implication: pq

Contrapositive: qp

Implication:If I am going to town, it is raining.

Contrapositive:

If it is not raining, then I am not going to town. The contrapositive has the same truth table as

the original implication.


19/48

Inverse of an Implication

Implication: pq

Inverse: pq

Implication:If I am going to town, it is raining.

Inverse:

If I am not going to town, then it is not raining. The inverse of an implication has the same

truth table as the converse of that implication.


20/48

Biconditional

if and only if, iff

pq

I am going to town if and only if it is raining. Bothpand qmust have the same truth value

for the assertion to be true.


21/48

Truth Table for (Biconditional)

p q p q

T T T

T F F

F T F

F F T


22/48

Truth Table Summary of Connectives

FF

T

Tp

FT

F

Tq p

qp

qp

qp

qp

q

TTFFFFTTTF

FFTTF

TTFTT


23/48

Compound Propositions

To construct complex truth tables, addintermediate columns for smaller (compound)

propositions

One column for each propositional variable

One column for each compound proposition

For npropositional variables there will be 2n

rows

Example:

(pq)r


24/48

Truth Tables for Compound Propositions

p q r p

q ~r (p

q) ~r

T T T T F F

T T F T T T

T F T T F F

T F F T T T

F T T T F F

F T F T T T

F F T F F T

F F F F T T


25/48

Precedence of Logical Operators

Parentheses gets the highest precedence

Then:

Operator Precedence 1

2

3

4

5


26/48

Precedence of Logical Operators

Examples:

p q r means (p q) r, not p (q r)

p

q

r means (p

q)

r


27/48

Translating English Sentences

To remove natural language ambiguity

Helps in reasoning

Examples:

You can access the Internet from campusonly if you are a computer science major oryou are not a freshman.

You cannot ride the roller coaster if you areunder 4 feet tall unless you are older than 16years old.


28/48

Logic and Bit Operations

Binary numbers use bits, or binary digits.

Each bit can have one of two values:

1 (which represents TRUE)

0 (which represents FALSE)

A variable whose value can be true or false is

called a Boolean variable. A Boolean variables

value can be represented using a single bit.


29/48

Logic and Bit Operations

Computer bit operations are exactly the same as the logic

operations we have just studied. Here is the truth table

for the bit operations OR, AND, and XOR:

x y x y x y x y

0 0 0 0 0

0 1 1 0 1

1 0 1 0 1

1 1 1 1 0


30/48

EECS 1590 -002

Discrete Structures

Chapter 1, Section 1.2

Propositional Logic applicationdemonstrated in class


31/48

EECS 1590 -002

Discrete Structures

Chapter 1, Section 1.3

Propositional Equivalences


32/48

Basic Terminology

A tautologyis a proposition which is always

true. Example:

pp

A contradictionis a proposition that is always

false. Example:

pp


33/48

Basic Terminology

p p p p p p

T F T F

F T T F


34/48

Basic Terminology

A contingencyis a proposition that is neither a

tautology nor a contradiction. Example:

pqr


35/48

Logical Equivalences

Two propositionspand qare logically

equivalent if they have the same truth values in

all possible cases.

Two propositionspand qare logically

equivalent ifpqis a tautology.

Notation: pq or pq


36/48

Determining Logical Equivalence

Consider the following two propositions:

(pq)

pq

Are they equivalent? Yes. (They are part of

DeMorgans Law, which we will see later.)

To show that (pq) and pqare

logically equivalent, you can use a truth table:


37/48

Equivalence of (pq) and pq

p q p q (p q) p q p q

T T T F F F F

T F T F F T F

F T T F T F F

F F F T T T T


38/48

Show that: p q pq

p q p p q p q

T T F T T

T F F F F

F T T T T

F F T T T


39/48

Determining Logical Equivalence

This is not a very efficient method. Why?

To be more efficient, we develop a series of

equivalences, and use them to prove otherequivalences.


40/48

Important Equivalences

IdentitypTp

pFp

Double Negation(p)p

DominationpTTpFF

Idempotentppp

p

p

p


41/48


Commutativepqqp

pqqp

Associative(pq) rp(qr)(pq) rp(qr)

Distributivep(qr)(pq) (pr)

p

(q

r)

(p

q)

(p

r)De Morgans(pq)pq

(pq)pq


42/48


Absorptionp(pq)p

p(pq)p

NegationppT

ppF


43/48

Example

Show that (pq) andpqare logicallyequivalent:

(pq) (p q) Example 3

(p) q 2ndDeMorgan lawp q double negation law

Q.E.D


44/48

Example

Show that

(p

(p

q)) andp

qarelogically equivalent:

(p(pq)) p(p q) 2ndDeMorgan

p[(p) q) 1stDeMorganp(p q) double negation

(pp) (p q) 2nddistributive

FALSE (p q) negation

(p q) FALSE commutativepq identity

Q.E.D


45/48

Equivalences Involving Implications

pq pq

pq qp

pq pq

pq (pq)

(pq) pq

(pq) (pr) p(q r)

(pr) (qr) (pq) r

(pq) (pr) p(q r)

(pr) (qr) (pq) r


46/48

Equivalences Involving Biconditionals

p q (p q) (q p)

p q p q

p q (p q) (p q)

(p q ) p q


47/48

Example

Show that (pq)(pq) is a tautology.

(p q)(p q) (p q) (p q) Example 3

(

p

q)

(p

q) 1

st

DeMorgan law p (q (p q)) Associative law

p ((p q) q) Commutative law

(p p) (q q) Associative law

TRUE TRUE Example 1

TRUE Domination law

QED


48/48

Steps to show logical equivalence

1. Use to simplify implication

pq pq

2. Use Distributiveand De Morgan's to simplify parenthesis

p(qr)(pq) (pr)

p(qr)(pq) (pr)

(pq)pq

(pq)pq

3. UseAssociativeto simplify parenthesis(pq) rp(qr)

(pq) rp(qr)

discreet structures ch1-1

Documents