01_ece math 311_propositional logic

8/12/2019 01_ECE MATH 311_Propositional Logic

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DISCRETE MATHEMATICS

ECE MATH 311

TOPIC 1:

PROPOSITIONAL LOGIC

By: Edison A. Roxas, ECE


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OBJECTIVES

Propositional Logic 2earoxas @ UST 2012

At the end of the topic, the students should be ableto:

1. Understand the role of Logic in DiscreteMathematics;

2. Determine if a given sentence is a propositionor not; and

3. Analyze problems involving unary logic and

connectives.4. Apply basic knowledge in Logic Circuit andSwitching Theory implementation.


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DISCRETE MATHEMATICS

It is the study of the structures which arediscrete rather than continuous.

Discrete objects are those which can beenumerated by integers, more formally,objects that can be characterized withcountable sets.

Discrete Mathematics finds use in computerscience, algorithm development,cryptography, research, and operations others.

earoxas @ UST 2012 Propositional Logic


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LOGIC

The study of reasoning.

Specifically concerned with whether reasoning

is correct.

Focuses on the relationship among statements

than the contents of particular statements.

Used in Mathematics to prove theorems.



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LOGIC

The rules of logic give precise meaning tomathematical statements.

These rules are used to distinguish between

valid and invalid mathematical arguments.The rules are used in the design of computer

programs; the verification of the correctness

of programs; and in many other ways.



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LOGIC

All Engineers are good in Math.

Anyone good in Math is a

Mathematician.

Therefore, all Engineers areMathematicians.



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PROPOSITIONS

It is a declarative sentence / an assertion (thatis, a sentence that declares a fact) that iseither TRUE or FALSE, but NOT BOTH.

It is the basic building block of any theory oflogic.

Letters are used to denote propositionalvariables (or statement variables), usually

propositional variables used are p, q, r and s.Any sentence associated with a TRUTH VALUE

is a proposition.



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PROPOSITIONS

The truth value of a proposition is T for TRUE andF for FALSE.

The area of logic that deals with the propositions

is called the propositional calculus orpropositional logic developed by the GreekPhilosopher Aristotle.

The methods of producing new propositions from

existing were discussed by English MathematicianGeorge Boole in his book, The Laws of Thought.



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Example 1.1: Which among the following

sentences are propositions?

1. The capital of Philippines is Manila.

2. 8 is a prime number.

3. There are 53 cards in a deck.

4. 2 + 5 = 75. 2 + 2 = 5

6. What is a Truth Table?

7. Answer the following questions completely.

8. x + 2 = 39. x y > 4

10. This statement is FALSE.



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TRUTH TABLE

A truth table is a list of possible combination

for individual propositions p1, p2, , pn.

To solve for the number of combinations

possible for any given value of propositions,

we use n = 2k.

where: k = number of columns

n = number of rows from 0 to 2K - 1



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LOGICAL CONNECTIVES

A propositional variables denotes an

arbitrary constant with unspecified truth value

P, Q, R

Basic Logical Connectives

Negation Not P ~P

Conjunction P and Q P Q

Disjunction P or Q P v Q

earoxas @ UST 2012 Propositional Logic 11


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NEGATION OPERATOR

The negation of p is denoted as

p not p.

Meaning, it is not the case of p.

/ ~ is a unary operator.

A unary operator assigns to each element in aset.

The truth tables of p and p are opposites.



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Example 1.2: Find the Negation of the

following proposition.

1. Today is Sunday.

2. CJ Corona is telling the truth.3. 2 + 3 = 19

4. The probability of an event to happen.

5. Binary 100112.



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CONNECTIVES

An assertion which contains at least one propositionalvariable is called a propositional form .

It is used to combine two propositions into a singleproposition.

CONJUNCTION: p q p and q

DISJUNCTION: p q p or q

Binary operators: and

Binary operators assigns to pair of elements in a set.



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CONJUNCTIONS ( )

The conjunction of p and q, denoted by

p q, is the proposition p and q.

The conjunction p q is TRUE when bothp and q are true; otherwise is FALSE.



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DISJUNCTION ( )

Disjunctions are Inclusive OR

Propositions, that is, at least one of the

propositions must be true for thepropositions to be TRUE.



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Example 1.3: Form the sentences required

by the following numbers.

1. If p: Math is Fun.

q: English is Hard.

What is the conjunction and disjunction?

2. If p: Algebra

q: Probabilities

Prerequisite of Discrete Math is the Disjunctions of p

and q.Prerequisite Advance Math is the Conjunctions of pand q.



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EXCLUSIVE OR

Exclusive OR Propositions are TRUE only when

exactly ONE of the constituent propositions is

TRUE.

Either This or That.

Entrees are served with soup or salad.

Exclusive OR Propositions are denoted as

p q.



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APPLICATION

LOGIC AND BIT OPERATIONS:

A bit (binary digit) is a symbol with twopossible values, namely zero (0) and one (1).

A True value can be represented by 1 and aFalse value by 0.

LOGIC CIRCUIT & SWITCHING THEORY DESIGN:

Logic Gates are electronic circuits used incombination to achieve a desirable output.



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APPLICATION

Example 1.4: Find the bitwise OR, bitwise AND, andbitwise XOR of the bit strings 011 101 101 and111 001 110.

Example 1.5: Use Logic Gates to implement thefollowing functions and find the possible outputs.

a. F = xy + xy

b. F = (x+y)(x+y)

c. F = xyz + xy + xyz

earoxas @ UST 2012 Propositional Logic 20

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