Note

1. LECTURE 3
Chapter 1.2
Propositional Equivalences
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University

2. Conditional Statement
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).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Truth Table for the Conditional Statement
P Q P → 𝑄
T T T
T F F
F T T
F F T

3. CONVERSE, CONTRAPOSITIVE,AND INVERSE
There are three related conditional statements 𝒑 → 𝒒 that
occur so often that they have special names.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Name Symbol Definition
CONVERSE 𝐪 → 𝒑 The proposition 𝐪 → 𝒑 is called the
converse of 𝐩 → 𝒒.
CONTRAPOSITIVE 𝒒 → 𝒑 The proposition 𝒒 → 𝒑 is
called the contrapositive of 𝐩 → 𝒒.
INVERSE 𝒑 → 𝑞 The proposition 𝒑 → 𝑞 is called
the inverse of 𝐩 → 𝒒

4. CONVERSE
Let p and q be propositions. The proposition 𝐪 → 𝒑 is called
the converse of 𝐩 → 𝒒.
The conditional statement q→ 𝒑 is false when q is true and p
is false, and true otherwise.
𝒒 → 𝒑 𝒎𝒆𝒂𝒏s "if q then p"
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Truth Table for the Conditional Statement (Converse)
Q P 𝐐 → 𝑷
T T T
T F F
F T T
F F T

5. CONTRAPOSITIVE
Let p and q be propositions. The proposition 𝒒 → 𝒑 is
called the contrapositive of 𝐩 → 𝒒.
𝒒 → 𝒑 𝒎𝒆𝒂𝒏s "if 𝒒 then 𝒑"
The conditional statement 𝒒 → 𝒑 is false when 𝒒 is true
and 𝒑 is false, and true otherwise.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Truth Table for the Conditional Statement (Contrapositive)
Q P 𝐐 𝐏 𝐐 → 𝐏
T T F F T
T F F T T
F T T F F
F F T T T

6. INVERSE
Let p and q be propositions. The proposition 𝒑 → 𝑞 is
called the inverse of 𝐩 → 𝒒
𝒑→ 𝑞 𝒎𝒆𝒂𝒏s "if 𝒑 then q"
The conditional statement 𝒑 → 𝑞 is false when 𝒑 is true
and 𝑞 is false, and true otherwise.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Truth Table for the Conditional Statement (Inverse)
P Q 𝐏 𝐐 𝑷 → 𝐐
T T F F T
T F F T T
F T T F F
F F T T T

7. Example 1: What are the contrapositive, the converse, and the inverse of the
conditional statement
"The home team wins whenever it is raining."
Solution: Because "q whenever p" is one of the ways to express the
conditional statement p → 𝑞 , the original statement can be rewritten as
"If it is raining, then the home team wins.“
p q
p= It is raining
q= The home team wins
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Name Symbol Convert Proposition into English Sentence
CONVERSE 𝐪 → 𝒑 If the home team wins, then it is raining.
𝐪 p
CONTRAPOSITIVE 𝒒 → 𝒑 If the home team does not win, then it is not raining.
𝒒 𝒑
INVERSE 𝒑 → 𝑞 If it is not raining, then the home team does not win.
𝒑 𝑞

8. p= It is raining
q= The home team wins
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
English Sentence Bangla Meaning
The home team wins whenever it is
raining.
যখনই বৃষ্টি হয় হহোম ষ্টিম জয়়ী হয়।
If it is raining, then the home team wins. যদি বৃষ্টি হয়, তোহলে হহোম ষ্টিম জজতলব।
If the home team wins, then it is raining.
CONVERSE
𝐪 → 𝒑
হহোম ষ্টিম জজতলে বৃষ্টি হয়।
If the home team does not win, then it is
not raining.
CONTRAPOSITIVE
𝒒 → 𝒑
হহোম ষ্টিম নো জজতলে বৃষ্টি হয় নো।
If it is not raining, then the home team
does not win.
INVERSE
𝒑 → 𝑞
বৃষ্টি নো হলে হহোম ষ্টিম জজতলব নো।

9. Tautology, Contradiction and Contingency
 A compound proposition that is always true, no matter what the
true values of the proposition that occur in it , is called tautology.
 A compound proposition that is always false is called
contradiction.
 A compound proposition that is neither a tautology nor a
contradiction is called a contingency.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University

10. Logical Equivalence
Compound proposition that have the same true values in all
possible cases are called logically equivalent. The compound
propositions p and q are called logically equivalent if p  q is a
tautology. The notation p ≡ q denotes that p and q are logically
equivalent.
Question: Show that ¬(𝑃˅𝑄)≡¬𝑃⋀¬Q
From the truth table we can say that, ¬(𝑃˅𝑄)≡ ¬𝑃⋀¬ Q
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University

11. Question: Show that ¬𝑷˅𝑸 ≡ 𝑷 → 𝑸
P Q ¬𝐏 ¬𝐏˅𝐐 𝐏 → 𝐐
T T F T T
T F F F F
F T T T T
F F T T T
From the truth table we can say that,
¬𝑷˅𝑸 ≡ 𝑷 → 𝑸

12. Question: Show that (P⋀𝑸) → (𝑷˅𝑸)is a tautology
P Q (P⋀𝐐) (𝐏˅𝐐) (P⋀𝐐) → (𝐏˅𝐐)
T T T T T
T F F T T
F T F T T
F F F F T
From the truth table we can say that, the true value of (P⋀𝑸) →
(𝑷˅𝑸) is all true so it is tautology.

13. Question: Check whether (P→ 𝑸)⋀(Q→ 𝑹) → (P → 𝑹)is tautologyor
not.
P Q R (P→ 𝑄) (P→ 𝑅) (𝑸 → 𝑅) (P→ 𝑄)⋀(Q→ 𝑅) (P→ 𝑄)⋀(Q→ 𝑅) → (P → 𝑅)
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F F T F T
F T T T T T T T
F T F T T F F T
F F T T T T T T
F F F T T T T T
From the truth table we can say that, the true value of (P⋀𝑄) →
(𝑃˅𝑄) is all true so it is tautology.

14. Precedence of Logical Operators
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Precedence of Logical Operators. PQ
Operator Operation Precedence
 Negation 1
 Conjunction 2
 Disjunction 3
→ Conditional 4
 Biconditional/ Bi-
implications
5

15. Question: How many rows appear in a truth table for
each of these compound propositions?
****Formula: Input=n, No. of rows=𝟐𝒏****
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Question
Solutions
Input No. of
Input
(n)
No. of
rows
(𝟐𝒏
)
No. of columns
1. P→ P P 1 𝟐𝟏
=2 3
Columns Heading: P, P, P→ P
2. (P ¬𝑄)→(𝑃 𝑄) P,Q 2 𝟐𝟐
=4 6
Columns Heading: P, Q, ¬𝑄, (P ¬𝑄), (𝑃 𝑄), (P ¬𝑄)→(𝑃 𝑄)
3. (P  R)T  (Q  T) P,Q,R,T 4 𝟐𝟒=16 8
Columns Heading: P,Q,R,T, (P  R) ,(P  R)T ), (Q  T), (P  R)T  (Q  T)
4. (P  R)  (Q 𝐒) P,Q,R,S 4 𝟐𝟒=16 9
Columns Heading: P,Q,R,S, R , 𝐒, (P  R) , (Q 𝐒) , (P  R)  (Q 𝐒)

16. Construct Truth Table For (𝑷¬𝑸) → (𝑷⋀𝑸)
Input Output
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
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
No. of Input=2, No. of rows=𝟐𝟐 = 𝟒
No. of Columns=6

17. Assignment 2
1.
2.
3.
4.
5.
6.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University