CMSC 56 | Lecture 1: Propositional Logic

Propositional Logic
Lecture 1, CMSC 56
Allyn Joy D. Calcaben

LOGIC
• the basis of all mathematical reasoning, and of all automated
reasoning.
• Rules of logic are used to distinguish between valid and
invalid mathematical arguments.

Propositions
• basic building blocks of logic
• A declarative sentence that is either true or false, but not
both.

Which are propositions?
Miagao is the capital of Iloilo Province.

What time is it?

What time is it?
2 + 5 = 9

What time is it?
2 + 5 = 9
x – 6 = 23

What time is it? A + B – C = D
2 + 5 = 9
x – 6 = 23

2 + 5 = 9 Read this carefully.
x – 6 = 23

2 + 5 = 9 Read this carefully.
x – 6 = 23 Miami is the capital of Florida

How do we represent propositions?

• We use letters to denote propositional variables (or
statement variables), that is , variables that represent
propositions, just as letter are used to denote numerical
variables.
• It uses conventional letters such as p, q, r, s, . . . .

• We use letters to denote propositional variables (or
statement variables), that is , variables that represent
propositions, just as letter are used to denote numerical
variables.
• It uses conventional letters such as p, q, r, s, . . . .
• Truth values is denoted by T, if it’s a true proposition, and
denoted by F, if it’s a false proposition.

• Area of logic that deals with propositions
• First developed systematically by the Greek Philosopher
Aristotle more than 2300 years ago.
Propositional Calculus /
Propositional Logic

• New propositions formed from existing propositions using
logical operators.
Compound Propositions

Definition 1
Let p be a proposition. The negation of p, denoted by
￢p (also denoted by p ), is the statement
“It is not the case that p.”
The truth value of the negation of p, ￢p, is the opposite of
the truth value of p.
Propositions

Find the negation of the proposition
“Michelle’s PC runs Linux”
and express this in simple English.
Example

Negation:
“It is not the case that Michelle’s PC runs Linux.”
more simply expressed negation:
“Michelle’s PC does not run Linux.”
Solution

Find the negation of the proposition
“Vanessa’s smartphone has at least 32GB of memory.”
and express this in simple English.
Example

Negation:
“It is not the case that Vanessa’s smartphone has at least
32GB of memory.”
Solution

Negation:
32GB of memory.”
can also be expressed as:
“Vanessa’s smartphone does not have at least 32GB of
memory.”
Solution

Negation:
32GB of memory.”
can also be expressed as:
“Vanessa’s smartphone does not have at least 32GB of
memory.”
even more simply as:
“Vanessa’s smartphone has less than 32GB of memory.”
Solution

Truth Table for the negation of a proposition p.
Truth Table
p ¬p
T
F

Truth Table
p ¬p
T F
F

Truth Table
p ¬p
T F
F T

Definition 2
Let p and q be propositions. The conjunction of p and q,
denoted by p ∧ q, is the proposition
“p and q.”
The conjunction p ∧ q is true when both p and q are true and
is false otherwise.
Propositions

Find the conjunction of the propositions p and q where p is
the proposition:
“Rachel’s PC has more than 500 GB free hard disk space.”
and q is the proposition:
“The processor in Rachel’s PC runs faster than 2 GHz.”
Example

The conjunction of these propositions, p ∧ q, is the
proposition:
“Rachel’s PC has more than 500 GB free hard disk space, and
the processor is Rachel’s PC runs faster than 2 GHz.”
Solution

This conjunction can be expressed more simply as:
“Rachel’s PC has more than 500 GB free hard disk space, and
its processor runs faster than 2 GHz.”
Solution

• Note that in logic the word “but” sometimes is used instead
of “and” in a conjunction.
• Example:
“The sun is shining, but it is raining.”
“The sun is shining and it is raining.”
Note

Truth Table for the Conjunction of Two Propositions.
Truth Table
p q p ∧ q
T T
T F
F T
F F

Truth Table
p q p ∧ q
T T T
T F
F T
F F

Truth Table
p q p ∧ q
T T T
T F F
F T
F F

Truth Table
p q p ∧ q
T T T
T F F
F T F
F F

Truth Table
p q p ∧ q
T T T
T F F
F T F
F F F

Definition 3
Let p and q be propositions. The disjunction of p and q,
denoted by p V q, is the proposition
“p or q.”
The disjunction p V q is false when both p and q are false and
is true otherwise.
Propositions

The disjunction of p and q, p V q, is the proposition:
“Rachel’s PC has at least 500 GB free hard disk space, or the
processor in Rachel’s PC runs faster than 2 GHz.”
Solution

Truth Table for the Disjunction of Two Propositions.
Truth Table
p q p V q
T T
T F
F T
F F

Truth Table
p q p V q
T T T
T F
F T
F F

Truth Table
p q p V q
T T T
T F T
F T
F F

Truth Table
p q p V q
T T T
T F T
F T T
F F

Truth Table
p q p V q
T T T
T F T
F T T
F F F

Definition 4
Let p and q be propositions. The exclusive or of p and q,
denoted by p ⊕ q, is the proposition that is true when exactly
one of p and q is true and is false otherwise.
Propositions

The exclusive or of p and q, p ⊕ q, is the proposition:
“Rachel’s PC has at least 500 GB free hard disk space, or the
processor is Rachel’s PC runs faster than 2 GHz, but not
both.”
Solution

Truth Table for the Exclusive or of Two Propositions.
Truth Table
p q p ⊕ q
T T
T F
F T
F F

Truth Table
p q p ⊕ q
T T F
T F
F T
F F

Truth Table
p q p ⊕ q
T T F
T F T
F T
F F

Truth Table
p q p ⊕ q
T T F
T F T
F T T
F F

Truth Table
p q p ⊕ q
T T F
T F T
F T T
F F F

Definition 5
Let p and q be propositions. The conditional statement p →
q, is the proposition “if p, then q.”
The conditional statement p → q is false when p is true and q
is false, and true otherwise.
In the conditional statement p → q, p is called the
hypothesis (or antecedent or premise) and q is called the
conclusion (or consequence).
Propositions

“if p, then q” “p implies q”
“if p, q” “p only if q”
“q if p” “q whenever p”
“q when p” “q is necessary for p”
“q follows from p” “p is sufficient for q”
“a necessary condition for p is q” “q unless ￢p”
“a sufficient condition for q is p”
p → q

Let p be the statement:
“Joy learns discrete mathematics.”
and q the statement:
“Joy will find a good job.”
Express the statement p → q as a statement in English.
Example

p → q is:
“If Joy learns discrete mathematics,
then she will find a good job.”
Solution

Express the statement p → q as a statement in English.
Given p is the proposition:
“I am elected.”
and q is the proposition:
“I will lower taxes.”
using “if p, then q”.
Example

p → q is:
“If I am elected, then I will lower taxes.”
Solution

Truth Table for the Conditional Statement p → q .
Truth Table
p q p → q
T T
T F
F T
F F

Truth Table
p q p → q
T T T
T F
F T
F F

Truth Table
p q p → q
T T T
T F F
F T
F F

Truth Table
p q p → q
T T T
T F F
F T T
F F

Truth Table
p q p → q
T T T
T F F
F T T
F F T

p → q is:
“If you get 100 on the exam, then I will give you 1.0.”
Another Example

Converse,
Contrapositive,
and Inverse

Given p → q
Converse: q → p
Contrapositive: ￢q → ￢p
Inverse: ￢p → ￢q
Propositions

Because “q wherever p” is one of the ways to express the
conditional statement p → q, the original statement,
“The home team wins whenever it is raining.”
can be rewritten as
“If it is raining, then the home team wins.”
Example

p → q: “If it is raining, then the home team wins.”
Converse: q → p
Solution

Converse: q → p
“If the home team wins, then it is raining.”
Solution

Converse: q → p
“If the home team does not win, then it is not raining.”
Solution

Converse: q → p
“If the home team does not win, then it is not raining.”
“If it is not raining, then the home team does not win.”
Solution

Which among the three
is equivalent to the
original statement?

Which among the three
is equivalent to the
original statement?
Answer: CONTRAPOSITIVE

Definition 6
Let p and q be propositions. The biconditional statement
p ↔ q, is the proposition “p if and only if q.”
The biconditional statement p ↔ q is true when p is and
q has the same truth values, and false otherwise.
Biconditional statements are also called bi–implications.
Propositions

There are some other common ways to express p ↔ q:
“p is necessary and sufficient for q”
“if p and q, and conversely”
“p iff q”
Propositions

Let p be the statement
“You can take the flight.”
and let q be the statement
“You buy a ticket.”
What is the p ↔ q statement using iff?
Example

Let p be the statement
“You can take the flight.”
and let q be the statement
“You buy a ticket.”
Then p ↔ q is the statement
“You can take the flight if and only if you buy a ticket.”
Example

Truth Table for the Biconditional Statement p ↔ q.
Note that p ↔ q has exactly the same truth value as
(p → q) ∧ (q → p)
Truth Table
p q p ↔ q
T T
T F
F T
F F

(p → q) ∧ (q → p)
Truth Table
p q p ↔ q
T T T
T F
F T
F F

(p → q) ∧ (q → p)
Truth Table
p q p ↔ q
T T T
T F F
F T
F F

(p → q) ∧ (q → p)
Truth Table
p q p ↔ q
T T T
T F F
F T F
F F

(p → q) ∧ (q → p)
Truth Table
p q p ↔ q
T T T
T F F
F T F
F F T

Truth Tables of
Compound Proposition

Precedence of Logical Operators
Operator Precedence
1
2
3
4
5

Precedence of Logical Operators
Operator Precedence
¬ 1
∧
∨
2
3
→
↔
4
5

Construct the truth table of the compound
proposition (p ∨ ￢q) → (p ∧ q)
p q ￢q p ∨ ￢q p ∧ q (p ∨ ￢q) → (p ∧ q)
T T
T F
F T
F F

Construct the truth table of the compound
proposition (p ∨ ￢q) → (p ∧ q)
p q ￢q p ∨ ￢q p ∧ q (p ∨ ￢q) → (p ∧ q)
T T F T T T
T F T T F F
F T F F F T
F F T T F F

Reference
CHAPTER 1. Rosen, Kenneth H.,Discrete Mathematics and Its
Applications, 7th edition, 2012.

CMSC 56 | Lecture 1: Propositional Logic

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