Short notes on Propositional Logic in Discrete Mathematics. Based on Discrete Mathematical Structure by Dr. Kamala Krithivasan Department of Computer Science and Engineering IIT Madras, NPTL Lecture. The next part will be uploaded soon.

1. PROPOSITIONAL LOGIC
Discrete Mathematics
JUNE 3, 2020
Avichal Vishnoi

2. Propositional Logic
[AUTHOR NAME] 1
Proposition Logic
An assertion is a statement. A ‘proposition’ is an assertion which is either
true or false but not both.
The following are propositions
a) 4 is a prime number.
b) 3 + 3 = 6.
c) The moon is made of cheese.
The following are not proposition
a) x + y > 4.
b) x = 3.
c) Are you leaving? (Question is not assertion)
d) Buy four books. (Order is not assertion)
e) This statement is false. (It is liar paradox. It is assertion but not
proposition).
A propositional variable denotes an arbitrary proposition with unspecified
truth value P, Q, R, etc.
Logical Connectives
❖ P and Q
❖ P or Q
❖ Not P
➢ P: John is 6’ tall.
➢ Q: There are 4 cows in the barn.
P and Q (P^Q): John is 6’ tall and there 4 cows in the barn.
P Q P ^ Q
0 0 0
0 1 0
1 0 0
1 1 1
P or Q (P ˅ Q): John is 6’ tall or there are 4 cows in the barn.

3. Propositional Logic
[AUTHOR NAME] 2
P Q P ˅ Q
0 0 0
0 1 1
1 0 1
1 1 1
Not P (¬ P / ~P):
P ¬ P/ ~P
0 1
1 0
An assertion which contains at least one propositional variable is called a
propositional form
And ^ conjunction
Or ˅ disjunction
Exclusive or (⊕): either P or Q is true
P Q P ⊕ Q
0 0 0
0 1 1
1 0 1
1 1 1
All these are ‘well-formed formula’ (wff) for propositional logic.

4. Propositional Logic
[AUTHOR NAME] 3
Implication
P implies Q …… P => Q
Notation:
• P ⊃ Q
• P ->Q
• P => Q
P ----, Premise, hypothesis, antecedent
Q ----, Conclusion, Consequence
P Q P => Q
0 0 1
0 1 1
1 0 0
1 1 1
It can be read as follows:
• If P, then Q
• P only if Q
• P is sufficient condition for Q
• Q is necessary condition for P
• Q if P
• Q follows from P
• Q provided P
• Q is a logical consequence of P
• Q whenever P
If P =>Q is a statement then
• Q =>P is the converse
• ¬ Q => ¬P is called contrapositive
P Q P=>Q ¬ 𝑷 ¬𝑷 ¬𝒒 => ¬𝑷
0 0 1 1 1 1
0 1 1 1 0 1
1 0 0 0 1 0
1 1 1 0 0 1

5. Propositional Logic
[AUTHOR NAME] 4
Note: If the statement is true then its converse may or may not be true but
its contrapositive will always true.
Note: If the statement is false then its converse may or may not be false but
its contrapositive will always false.

6. Propositional Logic
[AUTHOR NAME] 5
Equivalence ()
• P if and only if Q
• P is equivalent to Q
• P is a necessary and sufficient condition for Q
P Q PQ
0 0 1
0 1 0
1 0 0
1 1 1
Tautology: A tautology is a propositional form whose truth value is
true for all possible values of its propositional variable e.g. P ˅¬P
Contradiction/Absurdity: It is a propositional form which is always
false e.g. P^¬𝑃
Contingency: A propositional form which is neither a tautology form
which is neither a tautology nor a contradiction is called contingency.