Logic

Discrete Mathematics
Propositional Logic
Harshit Kumar

Slides borrowed from Wong Chung Hoi of CUHK

Agenda
• Proposition (Statement)
• Logic Operators
• Logical formula
• Problems
– Proofing formula
– Constructing formula from truth table
– Simplifying formula

Proposition (Statement)
• A sentence that is either TURE or FALSE
– 1 + 1 = 2.
– 1 + 1 = 3.
– Let’s end the tutorial now.
– This tutorial is boring.
– Wake up and listen to me!
– There are no aliens.
– x > 0.
– He is handsome. ?
• Tautology – proposition that is always true
• Contradiction – proposition that is always false

He has courage!
?
This man has courage!

You are doing it wrong!
?
Your way of pretending to be a penguin is wrong!

Logic Operators
• Let p and q be a proposition.
• Operators:
– Negation
– Conjunction
– Disjunction
– Conditional
– Bi-conditional

Negation (NOT)
• Negation (NOT)
– Flip the truth value.

• Example:
– p: My car is blue. ¬p: My car is not blue.
– p: Peter is good. ¬p: Peter is not good.
– p: 10 > 15. ¬p: 10 < 15 or 10 = 15

p: 49% different is a lot
¬p: 49% different is not a lot

p: Elephants are larger than the moon

¬p: Elephants are smaller than or equal size to the moon

Conjunction (AND)
• Conjunction (AND)
– True only when p and q are True

• Example:
– Quiz one is easy and quiz two is difficult.
– Peter is so handsome and smart.
– Peter is so handsome and Peter is so smart.

Disjunction (OR)
• Disjunction (OR)
– True when either p or q or both are true.

• Example
– I will go with my sister or I will go with my brother.

Exclusive Or (XOR)
• Exclusive Or (XOR)
– True only when either p or q is true but not both

• Example
– Tomorrow is Thursday or tomorrow is Friday.

Conditional (If … then …)
• Conditional (If… then…)
– “If p then q” can only be disproved to be false when p
really happens but q doesn’t.
– p is sufficient condition q.
– q is necessary condition p.
– “p if q” = “if q then p”
– “p only if q” = “if p then q”

• Example
– If tomorrow is hot, I will go swimming.
(If tomorrow is cold, you can’t disprove the statement.)

Bi-Conditional (If and only if)
• Bi-Conditional (If and only if)
– “p if and only if q” can only be disproved when p
happens but not q or vice versa.
– p (q) is necessary and sufficient condition for q (p)
–

• Example:
– A computer program is correct
if and only if it produces correct
answer for all possible sets of
input data

Logical Formula
• Distribution Laws:

• De Morgan’s Laws:

• Absorption Laws:

Proofing logical equivalent 1
• By truth table
• E.g. Show that De Morgan’s law

Proofing logical equivalent 2
• By logical rules
• E.g. Show that

De Morgan’s Law

De Morgan’s Law

Constructing Formula 1
• By using only
• Find the logical formula for
1. Truth table
2. When will this formula
be True?
3. Simplify

• Exercise: Try to construct an logical formula for
, ,

1. Truth table
2. When will this formula be True?
3. Simplify

• Exercise: Verify the above
formula.

1. Truth table
2. When is this formula True?

3. Simplify

De Morgan’s law

Distribution Laws

Distribution Laws

Distribution Laws

De Morgan’s law

Simplifying Formula
• Simplify
De Morgan’s law

Distribution Laws

Summary
• What is proposition?
• Common logical operator.
• Proving Equivalent of formula.
• Constructing formula from truth table.
• Simplifying formula.

Logic

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