ICS 253: Discrete Structures I

ICS 253: Discrete Structures I

Dr. Nasir Al-DarwishComputer Science Department King Fahd University of Petroleum and [email protected]

Spring Semester 2014 (2013-2)

Propositional LogicSection 1.1

KFUPM: Dr. Al-Darwish © 2014 2

Grading Lecture attendance: -1% per 2 unexcused

absences

Assignments & Quizzes: 20%

Two Major Exams: 25% per exam

Final Exam: 30%


Expectations This is really a fun course! The course covers some of the most useful math

you’ll ever learn. It teaches abstraction, describe (or model) a

given problem using precise formal notation

Hints for success Read the textbook.

Lectures really do help!

Do the homework.


Traditionally, logic distinguishes between valid and invalid statements (2-valued logic). That is, a proposition (claim) is either true or false.

There are other types of logic, e.g. fuzzy logic

The building blocks of logic are propositions.

A proposition (claim) is a declarative statement that is either true or false.

Propositional Logic


What’s a proposition?

Propositions Not Propositions

3 + 2 = 32 Bring me coffee!

Math is my favorite subject.

3 + 2

Every cow has 4 legs. Do you like Cake?

There is other life in the universe.

I will go fishing tomorrow.

X – 2 = 10 (depends on X)

A proposition is a declarative statement that’s either TRUE or FALSE (but not both, not conditional).

Propositions versus Not Propositions


Atomic versus Compound Propositions

Logical Operation Symbol

Negation (NOT)

Conjunction (AND)

Disjunction (OR)

Exclusive OR

Implication (imply)

Biconditional (2-way imply)

A proposition is either atomic (simple) or compound

A compound proposition is formed by combining atomic propositions using logical operations (operators)


Logical Operations

Let p & q be propositions, then the following are compound propositions:

Negation: p = not p

Conjunction: p q = p AND q

Disjunction: p q = p OR q

Exclusive OR: p q = p XOR q

Implication: p q = if p then q

Biconditional: p q = p if and only if q


Precedence of Logical Operators

In a parenthesized expression, the inner most parenthesized part is computed first

Binary operators with same precedence are applied left-to-right; unary are applied right-to-left

The following is the order of operators by precedence

, , v , , This means for example (assuming parenthesis are not

present)

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


Truth Tables

A truth table shows the truth values of a compound proposition in relation to the truth values of its constituent propositions

Normally, the truth table will have as many rows as the different possible truth value of its constituent propositions. For example, a compound proposition involving two simple propositions P, Q will have 4 rows(4 different T/F combinations for P and Q).


Propositional Logic: Negation

Suppose p is a proposition. The negation of p is written p and has

meaning:

“It is not the case that p.”

Example: P = Pizza is my favorite food.

P = It is not the case that pizza is my favorite food. More simply, pizza is NOT my favorite food.

Truth table for negation

p p

FT

TF

Note that p is a

proposition!


Propositional Logic: Conjunction

Conjunction corresponds to English “and” p q is true exactly when p is true and q is true

(or both are true).

Example: Amy is curious AND clever.

Truth table for conjunction

p q p q

FFTT

FTFT

FFFT


Propositional Logic: Disjunction

Disjunction corresponds to English “or” p q is true when p is true or q is true (or

both are true).

Example: Michael is brave OR nuts.

Truth table for disjunction

p q p q

FFTT

FTFT

FTTT


Propositional Logic: Implication

The implication p q corresponds to English: “if p then q” or “p implies q”

P Q can be considered as a contract that relates a condition P to a conclusion Q P = build me a house, Q = pay you $1 million

If you build me a house then I will pay you $1 million

The contract is violated if the house is built but the payment is not made

p q is false only in the case where p is true and q is false

If p then 2+2=4. (This is true. Why?)

Truth table for implication

p q p q

FFTT

FTFT

TTFT


Logical Equivalence

How many different logical binary operations could we define?

To answer, we need the notion of “logical equivalence.”

16

How many different logical operations do we need?

??

p is logically equivalent to q if their truth tables are the same. Logical equivalence is denoted by p q (or p q).


Logical Equivalence, cont.

Challenge: Try to find a proposition that is equivalent to p q, but that uses only the connectives , , and .

p q p q

FFTT

FTFT

TTFT

p q p p q

FFTT

FTFT

TTFF

TTFT


Propositional Logic, a proof of one famous

Distributivity: p (q r) (p q) (p r)

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

F F F F F F F F

F F T F F F T F

F T F F F T F F

F T T T T T T T

T F F F T T T T

T F T F T T T T

T T F F T T T T

T T T T T T T T

The above truth table is a “proof” of “law of distributivity for over

.” Note: In logic, distributes over . Also, distributes over .


Expressing Implications

The implication p q essentially says that we cannot have p true yet q is false, (p q).

Thus, p q (p q) (1) . (The equivalence can be shown by truth table)

Simplifying the RHS (use DeMorgan’s law), gives p q p q (2)

It is important to observe that p q also means (because we cannot have p and not q): not q not p

It is easy to show that p q q p (3)

This last equivalence is known as the ContraPositive (negative of positive) .

I liken it to “the cub is half full the cub is half empty”


Expressing Implications, cont.

Definition: p is sufficient for q if whenever p is true, q is true.

Thus, p q means that p is sufficient for q.

Definition: p is necessary for q if whenever p is false, q is false.

Thus, p q means that q is necessary for p. (Recall from previous slide that p q q p)



The following are some of the ways of expressing p q

p implies q if p then q if p, q q if p (stating the conclusion first) p only if q* (see next two slides for justification) p is sufficient for q q is necessary for p q whenever (when) p q follows from p q unless ¬p*

** These seem confusing



p q can be expressed as:

p only if q: p cannot be true if q is not true This corresponds to the contrapositive of p q

q unless p If p is false, then q must be true Example:

If You fail the final exam then you will get F You will get F unless you do not fail the final exam


Propositional Logic: Biconditional

The biconditional proposition p ↔ q (read as, p if and only if q) states that p and q are always equal (either they are both true or both false)

The biconditional proposition p ↔ q is equivalent to “(p q) and (q p)”

Because in the expression “p if and only if q” , “p if q” is q p,

it follows that p q is “p only if q”

Question: Is p ↔ q (p q) (p q)?

p q p ↔ q

FFTT

FTFT

TFFT

Truth table for biconditional


Propositional Logic, Some Definitions

Contrapositive: The Contrapositive of p q is q p

Example: "If it is noon, then I am hungry."

Contrapositive: “If I am not hungry, then it is not noon.”

Note: This is a must-remember because an implication is logically equivalent to its contrapositive.

Converse: The Converse of p q is q p

Example: "If it is noon, then I am hungry."

Converse: "If I am hungry, then it is noon."

Inverse: The Inverse of p q is p q

Example: "If it is noon, then I am hungry. "

Inverse: "If it is not noon, then I am not hungry."


Propositional Logic, More Definitions

A tautology is a proposition that’s always TRUE.

A contradiction is a proposition that’s always FALSE.

p p p p p p

F T T F

T F T F


Translating English Sentences

English (and every other human language) is often ambiguous. Translating sentences into compound propositions removes the ambiguity.

Question: How can this English sentence be translated into a logical expression?

"You can access the Internet from campus only if you are a computer science major or you are not a freshman. "

Solution: Let a, c, and f, respectively, denote "You can access the Internet from campus", "You are a computer science major", and "You are a freshman"

The sentence can be translated into:

a → (c ν ¬f).


Example: How can this English sentence be translated into a logical expression?

"You cannot enter into the pool if you are under 4 feet

tall unless you are older than 16 years old."Solution: Let p, r, and s, respectively, denote "You can enter into the pool" , "You are under 4 feet tall", and "You are older than 16 years old" The sentence can be translated into:

(r Λ ¬ s) → ¬p.

Translating English Sentences


We will be using propositional logic as a foundation for formal proofs. Later we will study different proof methods; these are needed for developing (or ensuring correctness) of algorithms.

Propositional logic is a key to writing correct program code. You can’t do any kind of conditional (if) statement without understanding the condition you’re testing. We gives examples of this in next lecture.

The logical operations we’ve discussed are also found in hardware and are called "logic gates" .

Why Study Propositional Logic?

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ICS 253: Discrete Structures I