Week 1: Introduction & Logic
MATHEMATICAL LOGIC
What is a Proposition?
In mathematics, we deal with statements that are either True or False. These are called propositions.
Definition
A proposition is a declarative statement that is either true or false, but not both.
Key Concept
Examples
Valid Propositions
- The sun rises in the East. (True)
- 2 + 2 = 5. (False)
- New Delhi is the capital of India. (True)
Not Propositions
- What is your name? (Question)
- Close the door. (Command)
- x + 2 = 5. (Depends on x)
Logical Connectives
We can combine propositions using logical connectives.
AND, OR, NOT
✨ Truth Tables
Let and be two propositions.
| (AND) | (OR) | (NOT) | ||
|---|---|---|---|---|
| T | T | T | T | F |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
Implication ()
This is one of the most important connectives. It is read as “if , then ”.
IF P THEN Q
🔥 The Promise Analogy
Think of as a promise.
- If I promise () to give you candy, and I give you candy (), I kept my promise (True).
- If I promise (), but don’t give you candy (not ), I broke my promise (False).
- If I didn’t promise (not ), whether I give you candy or not, I didn’t break any promise (True).