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.
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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
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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
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