CSCI 220 | Fall 2025
Discrete Structure
Propositional Logic
Discrete Mathematics and its Application
Section 1.1, 1.2, 1.3
Definition of Proposition
A proposition is a declarative sentence (that is, a sentence that declares a
fact) that is either always true or always false, but not both.
Definition of Proposition
A proposition is a declarative sentence (that is, a sentence that
declares a fact) that is either always true or always false, but not both.
Examples:
• 2 is the only even prime. Yes, true prop
• 1 + 1 = 2.
Yes, true prop
• 2 + 2 = 5.
Yes, false prop
• What time is it?
No
• Read this carefully.
no
• 𝑥+1=2
yes
Exercise
Which of these sentences are propositions?
1) 0 is a positive integer.
Yes, False Prop
2) Today is Thursday. No, We don’t know what the date is today
no, n is not defined
3) 𝑛 > 7.
4) Every even number and be written as 2 times an integer.
Yes, true
Definition of Proposition
A proposition is a declarative sentence (that is, a sentence that
declares a fact) that is either always true or always false, but not both.
We use letters to denote propositional variables, that is, variables that
represent propositions. The truth value of a proposition is true, T, if it
is a true proposition, and false, F, if it is a false proposition.
Examples:
• 2 is the only even prime.
• 2+2=5
P=T
q=F
Negation of Proposition
Negation of Proposition
Examples:
• Find the negation of the proposition
“This flash drive has at least 32 GB of memory”
and express this in simple English.
This flash drive has less than 32 GB of memory.
Exercise
Find the negation of the proposition
“Every student enrolling this course are computer science major.”
and express this in simple English.
not every student ….. are CS majors
Conjunction of Proposition
Conjunction of Proposition
Examples:
Let p the propositions that state “Alex is a computer
science major” and q be “Alex is a math major”.
Find the conjunction of the propositions p and q.
Disjunction of Proposition
Disjunction of Proposition
Examples:
Let p the propositions that state “Alex is a computer
science major” and q be “Alex is a math major”.
Find the disjunction of the propositions p and q.
Exclusive OR of Proposition
Exclusive OR of Proposition
Examples:
Let p the propositions that state “Alex is a computer
science major” and q be “Alex is a math major”.
Find the exclusive or of the propositions p and q.
Conditional Statements
Conditional Statements
Examples:
Let p the propositions that state “You pass all the
exams” and q be “you pass this course”.
Express the statement 𝑝 → 𝑞 as a statement in
English.
Conditional Statements
Examples:
If you pass all the exams, then you will pass this
course.
Converse, Contrapositive, and Inverse
Given a conditional statement 𝑝 → 𝑞.
The proposition 𝑞 → 𝑝 is called the converse of 𝑝 → 𝑞.
The proposition ¬𝑞 → ¬𝑝 is called the contrapositive of 𝑝 → 𝑞.
The proposition ¬𝑝 → ¬𝑞 is called the inverse of 𝑝 → 𝑞.
Converse, Contrapositive, and Inverse
Given a conditional statement 𝑝 → 𝑞.
The proposition 𝑞 → 𝑝 is called the converse of 𝑝 → 𝑞.
The proposition ¬𝑞 → ¬𝑝 is called the contrapositive of 𝑝 → 𝑞.
The proposition ¬𝑝 → ¬𝑞 is called the inverse of 𝑝 → 𝑞.
The negation of 𝑝 → 𝑞 is 𝑝 ∧ ¬𝑞.
Expressions of Conditional Statement
Given a conditional statement 𝑝 → 𝑞. It can be expressed as the
following:
Biconditional Statements
Examples:
Let p the propositions that state “You pass all the
exams” and q be “you pass this course”.
Express the biconditional statement 𝑝 𝑞 as a
statement in English.
Biconditional Statements
Examples:
You pass all the exams if and only if you pass this
course.
Expressions of Biconditional Statement
Given a conditional statement 𝑝
following:
𝑞. It can be expressed as the
Summary
Truth Tables of Compound Propositions
Construct the truth table of the compound proposition
Precedence of Logical Operators
Truth Tables of Compound Propositions
Construct the truth table of the compound proposition
• Make your own proposition.
Logic and Bit Operations
Logic and Bit Operations
Logic and Bit Operations
Example: 101010011 is a bit string of length nine.
Logic and Bit Operations
Example: 101010011 is a bit string of length nine.
Exercise: Find the bitwise OR, bitwise AND, and bitwise XOR of the bit
strings 01 1011 0110 and 11 0001 1101.
Definitions of Tautology,
Contradiction, and Contingency
Definitions of Tautology,
Contradiction, and Contingency
Definition of Logical Equivalent
Definition of Logical Equivalent
In other words, compound propositions that have the same truth values in all
possible cases are called logically equivalent.
Definition of Logical Equivalent
In other words, compound propositions that have the same truth values in all
possible cases are called logically equivalent.
Exercise: Show that ¬(𝑝 ∨ 𝑞) and ¬𝑝 ∧ ¬𝑞 are logically equivalent.
Definition of Logical Equivalent
In other words, compound propositions that have the same truth values in all
possible cases are called logically equivalent.
Exercise: Show that ¬(𝑝 ∨ 𝑞) and ¬𝑝 ∧ ¬𝑞 are logically equivalent.
Logical Equivalent Exercise
Show that 𝑝 → 𝑞 and ¬𝑝 ∨ 𝑞 are logically equivalent. (This is known as
the conditional-disjunction equivalence.)
Logical Equivalent Exercise
Show that 𝑝 → 𝑞 and ¬𝑝 ∨ 𝑞 are logically equivalent. (This is known as
the conditional-disjunction equivalence.)
Logical Equivalent Exercise
Show that 𝑝 ∨ (𝑞 ∧ 𝑟) and (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) are logically equivalent. This
is the distributive law of disjunction over conjunction.
Logical Equivalent Exercise
Show that 𝑝 ∨ (𝑞 ∧ 𝑟) and (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) are logically equivalent. This
is the distributive law of disjunction over conjunction.
Logical Equivalences
Logical Equivalences Involving
Conditional Statements
Logical Equivalences Involving
Biconditional Statements
Constructing New Logical Equivalences
Show that ¬(𝑝 → 𝑞) and 𝑝 ∧ ¬𝑞 are logically equivalent.
Constructing New Logical Equivalences
Show that ¬(𝑝 ∨ (¬𝑝 ∧ 𝑞)) and ¬𝑝 ∧ ¬𝑞 are logically equivalent by
developing a series of logical equivalences.
Constructing New Logical Equivalences
Show that (𝑝 ∧ 𝑞) → (𝑝 ∨ 𝑞) is a tautology.
. . . END