cs203 discrete mathematical structures logic proposition logic

CS203Discrete Mathematical Structures

Logic Proposition logic

What is logic?

The branch of philosophy concerned with analysing the patterns of reasoning by which a conclusion is drawn from a set of premises, without reference to meaning or context

(Collins English Dictionary)

Digital Logic

Propositional logic

• Simple types of statements, called propositions, are treated as atomic building blocks for more complex statements

1) Alexandria is a port or a holiday resort.

2) Alexandria is not a port.

Therefore, Alexandria is a holiday resort

Basic connectives and truth tables

statements (propositions): declarative sentences that areeither true or false--but not both.

Eg. Ahmed Hassan wrote Gone with the Wind. 2+3=5.

not statements:

What a beautiful morning!Get up and do your exercises.

Statement (Proposition)

A Statement is a sentence that is either True or False

Examples:

Non-examples:

x+y>0

x2+y2=z2

True

False2 + 2 = 4

3 x 3 = 8

787009911 is a prime

They are true for some values of x and y

but are false for some other values of x and y.

Today is Tuesday.

Propositional connectives

• Propositional logic has four connectives

Name Read as Symbol

negation ‘not’

conjunction ‘and’

disjunction ‘or’

implication ‘if…then…’

Logical Operators

• About a dozen logical operators– Similar to algebraic operators + * - /

• In the following examples,– p = “Today is Friday”

– q = “Today is my birthday”

Logical operators: Not

• A “not” operation switches (negates) the

truth value

• Symbol: or ~

p = “Today is not Friday”

p p

T F

F T

Logical operators: And

• An “and” operation is true if both operands are true

• Symbol: – It’s like the ‘A’ in And

• pq = “Today is Friday and today is my birthday”

p q pq

T T T

T F F

F T F

F F F

Logical operators: Or

• An “or” operation is true if either

operands are true

• Symbol:

• pq = “Today is Friday or

today is my birthday (or

possibly both)”

p q pq

T T T

T F T

F T T

F F F

Logical operators: Conditional

• A conditional means “if p then q”

• Symbol: • pq = “If today is

Friday, then today is my birthday”

• p→q=¬pq

p q pq

T T T

T F F

F T T

F F T

Logical operators: Bi-conditional

• A bi-conditional means “p if and only if q”• Symbol: • Alternatively, it means

“(if p then q) and (if q then p)”

• Note that a bi-conditional has the opposite truth values of the exclusive or

p q pq

T T T

T F F

F T F

F F T

Boolean operators summary

• Learn what they mean, don’t just memorize the table!

not not and or xor conditional Bi-conditional

p q p q pq pq pq pq pq

T T F F T T F T T

T F F T F T T F F

F T T F F T T T F

F F T T F F F T T

Precedence of operators

• Just as in algebra, operators have precedence– 4+3*2 = 4+(3*2), not (4+3)*2

• Precedence order (from highest to lowest): ¬ → ↔– The first three are the most important

• This means that p q ¬r → s ↔ t yields: (p (q (¬r)) → s) ↔ (t)

• Not is always performed before any other operation

Example

s: Aya goes out for a walk.t: The moon is out.u: It is snowing.

( )t u s : If the moon is out and it is not snowing, thenAya goes out for a walk.

If it is snowing and the moon is not out, then Aya will not go out for a walk. ( )u t s

Translating English Sentences– p = “It is below freezing”– q = “It is snowing”

• It is below freezing and it is snowing• It is below freezing but not snowing• It is not below freezing and it is not snowing• It is either snowing or below freezing (or both)• If it is below freezing, it is also snowing• It is either below freezing or it is snowing,

but it is not snowing if it is below freezing• That it is below freezing is necessary and

sufficient for it to be snowing

pqp¬q¬p¬qpqp→q

((pq)¬(pq))(p→¬q)

p↔q

Logical Equivalence

0011

0101

1100

1101

1101

p q p p q p q

s s1 2

L3 19

Tables of Logical Equivalences

• Identity lawsLike adding 0

• Domination lawsLike multiplying by 0

• Idempotent lawsDelete redundancies

• Double negation“I don’t like you, not”

• Commutativity Like “x+y = y+x”

• AssociativityLike “(x+y)+z = y+(x+z)”

• DistributivityLike “(x+y)z = xz+yz”

• De Morgan

Tables of Logical Equivalences

• Excluded middle• Negating creates opposite• Definition of implication in

terms of Not and Or

Fundamentals of Logic

A compound statement is called a tautology(T0) if it is true for all truth value assignments for its component statements. If a compound statement is false for all such assignments, then it is called a contradiction(F0).

p p q

p p q

( )

( )

: tautology

: contradiction

Propositional Logic - 2 more defn…A tautology is a proposition that’s always TRUE.

A contradiction is a proposition that’s always FALSE.

p p p p p p

T F

F T

T

T

F

F

23

Tautology example

Demonstrate that

[¬p (p q )]q

is a tautology in two ways:

1. Using a truth table – show that [¬p (p q )]q is always true

2. Using a proof (will get to this later).

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T

T F

F T

F F

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F

T F F

F T T

F F T

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T

T F F T

F T T T

F F T F

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F

T F F T F

F T T T T

F F T F F

Tautology by truth table

p q ¬p p q ¬p (p q ) [¬p (p q )]q

T T F T F T

T F F T F T

F T T T T T

F F T F F T

29

Tautologies, contradictions and programming

Tautologies and contradictions in your code usually correspond to poor programming design. EG:

•while(x <= 3 || x > 3)

x++;•if(x > y)

if(x == y)

return “never got here”;

dual

Let s be a statement. If s contains no logical connectivesother than and , then the dual of s, denoted sd, is thestatement obtained from s by replacing each occurrence ofand by and , respectively, and each occurrence of Tand F by F and T, respectively.

Eg. )()(:),()(: FrqpsTrqps d

The dual of is p q ( ) p q p qd

The Principle of Duality

Theorem () Let s and t be statements.If , then .s t s td d

Ex. P p q p q: ( ) ( ) is a tautology. Replace

each occurrence of p by r s

P r s q r s q1: [( ) ] [ ( ) ] is also a tautology.

First Substitution Rule (replace each p by another statement q)

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Converse

The converse of a logical implication is the reversal of the implication. I.e. the converse of p q is q p.

EG: The converse of “If Donald is a duck then Donald is a bird.” is “If Donald is a bird then Donald is a duck.”

p q and q p are not logically equivalent.

Converse

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

Converse

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

TTFF

TFTF

Logical Non-Equivalence of Conditional and Converse

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

TTFF

TFTF

TFTT

Logical Non-Equivalence of Conditional and Converse

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

TTFF

TFTF

TFTT

TTFT

Logical Non-Equivalence of Conditional and Converse

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

TTFF

TFTF

TFTT

TTFT

TFFT

Fundamentals of Logic

Compare the efficiency of two program segments.z:=4;for i:=1 to 10 do begin x:=z-1; y:=z+3*i; if ((x>0) and (y>0)) then writeln(‘The value of the sum x+y is’, x+y) end

.

.

.if x>0 then if y>0 then …

rqp )( )( rqp

Number of comparisons?20 vs. 10+3=13

logically equivalent

Derivational Proof Techniques

When compound propositions involve more and more atomic components, the size of the truth table for the compound propositions increases

Q1: How many rows are required to construct the truth-table of:( (q(pr )) ((sr)t) ) (qr )

Q2: How many rows are required to construct the truth-table of a proposition involving n atomic components?

Derivational Proof Techniques

A1: 32 rows, each additional variable doubles the number of rows

A2: In general, 2n rows

Therefore, as compound propositions grow in complexity, truth tables become more and more unwieldy. Checking for tautologies/logical equivalences of complex propositions can become a chore, especially if the problem is obvious.

L3 41

Derivational Proof Techniques

EG: consider the compound proposition

(p p ) ((sr)t) ) (qr )

Q: Why is this a tautology?

L3 42

Derivational Proof Techniques

A: Part of it is a tautology (p p ) and the disjunction of True with any other compound proposition is still True:

(p p ) ((sr)t )) (qr ) T ((sr)t )) (qr ) TDerivational techniques formalize the

intuition of this example.

Propositional Logic - an unfamous

(p q) q p q

if NOT (blue AND NOT red) OR red then…

(p q) q

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

DeMorgan’s

Double negationAssociativity

Idempotent

L3 44

Tautology by proof[¬p (p q )]q

L3 45

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

L3 46

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE

L3 47

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

L3 48

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

L3 49

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

L3 50

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

[p ¬q ] q Double Negation

L3 51

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

[p ¬q ] q Double Negation

p [¬q q ] Associative

L3 52

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

[p ¬q ] q Double Negation

p [¬q q ] Associative

p [q ¬q ] Commutative

L3 53

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

[p ¬q ] q Double Negation

p [¬q q ] Associative

p [q ¬q ] Commutative p T ULE

L3 54

Tautology by proof[¬p (p q )]q

[(¬p p)(¬p q)]q Distributive

[ F (¬p q)]q ULE [¬p q ]q Identity

¬ [¬p q ] q ULE

[¬(¬p) ¬q ] q DeMorgan

[p ¬q ] q Double Negation

p [¬q q ] Associative

p [q ¬q ] Commutative p T ULE T Domination

Writing Logical Formula for a Truth Table

Digital logic:

Given a digital circuit, we can construct the truth table.

Now, suppose we are given only the truth table (i.e. the specification,

e.g. the specification of the majority function), how can

we construct a digital circuit (i.e. formula) using only

simple gates (such as AND, OR, NOT)

that has the same function?

Examples

1. “I don’t study well and fail” is logically equivalent to “If I study well, then I don’t fail”

2. Write a C program that represents the compound proposition (pq)r

Decide whether the following statements are tautologies or contradictions or neither

1. (p q) (q p).

2. (p q) (q ¬p).

3. p ¬p.

4. (p ¬q) (q ¬p).

Use truth table to find

• - P Q P R• P -Q R P R - Q

• A B -C D E F

• (P ↔ Q) (P˅ Q)

• [P ˄ (P Q)] Q

Which of the following equivalent to -(P˄Q)

• (-p) ˄ (-Q)

• P ˅ Q

• (-P) ˄ Q

• (-P) ˅ (-Q)