🔗 Combining Statements
Just as arithmetic has operators like + and × to combine numbers, logic has logical operators to combine statements. The three fundamental operators are:
- AND — conjunction (symbol: ∧)
- OR — disjunction (symbol: ∨)
- NOT — negation (symbol: ¬)
We'll use two statements throughout this lesson:
p = "It is raining." | q = "I have an umbrella."
🤝 AND (Conjunction) p ∧ q
A conjunction is true only when both statements are true. If even one is false, the whole conjunction is false.
Example
p ∧ q: "It is raining AND I have an umbrella."
Both must be true at the same time for this to be true.
"I like pizza and I like ice cream." — True only if you like both.
| p | q | p ∧ q (AND) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
💡 Memory Tip
AND is like a strict teacher — both students must raise their hand, or the answer is no!
🔀 OR (Disjunction) p ∨ q
A disjunction is true when at least one statement is true. It is only false when both are false.
Example
p ∨ q: "It is raining OR I have an umbrella."
This is true even if only one of those things is true.
"You can have cake OR pie." — In logic, you could have both (inclusive OR).
| p | q | p ∨ q (OR) |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Inclusive OR vs. Exclusive OR (XOR):
Logic's OR is inclusive — it allows both to be true. Everyday English sometimes uses exclusive or ("you can have coffee or tea" often implies not both). In logic, OR always means "at least one."
💡 Memory Tip
OR is like a lenient teacher — even one student raising their hand is enough!
🔁 NOT (Negation) ¬p
You already saw negation in the previous lesson! NOT simply flips a statement's truth value.
Example
¬p: "It is not raining."
¬q: "I do not have an umbrella."
| p | ¬p (NOT p) |
|---|---|
| T | F |
| F | T |
🔢 Combining Operators
You can nest operators together to create complex logical expressions — just like using parentheses in math.
Example: ¬p ∧ q
"It is not raining and I have an umbrella."
| p | q | ¬p | ¬p ∧ q |
|---|---|---|---|
| T | T | F | F |
| T | F | F | F |
| F | T | T | T |
| F | F | T | F |
Example: ¬(p ∨ q) — De Morgan's Law
The negation of "p OR q" equals "NOT p AND NOT q" — one of De Morgan's famous laws:
¬(p ∨ q) ≡ ¬p ∧ ¬q
"It is NOT the case that it's raining or I have an umbrella" = "It's not raining AND I don't have an umbrella."
¬(p ∧ q) ≡ ¬p ∨ ¬q
"It is NOT the case that it's BOTH raining AND I have an umbrella" = "It's not raining OR I don't have an umbrella."
✏️ Practice — Click to Reveal Answers
Let p = "The light is on." (True) and q = "The door is open." (False).
1. What is the truth value of p ∧ q?
FALSE. AND requires both to be true. Since q is false, p ∧ q is false.
2. What is the truth value of p ∨ q?
TRUE. OR only needs at least one true. Since p is true, p ∨ q is true.
3. What is the truth value of ¬p ∨ ¬q?
TRUE. ¬p = F, ¬q = T. F ∨ T = TRUE.
4. What is the truth value of ¬(p ∧ q)?
p ∧ q = F (from question 1). So ¬(p ∧ q) = TRUE.
By De Morgan: ¬(p ∧ q) = ¬p ∨ ¬q = F ∨ T = TRUE ✓
5. Write a real-world sentence for: ¬p ∧ q
"The light is NOT on AND the door is open."
Truth value: ¬p = F, so F ∧ F = FALSE.
🎯 Key Takeaways
- AND (∧) — true only when both statements are true.
- OR (∨) — true when at least one statement is true.
- NOT (¬) — flips the truth value of a statement.
- Operators can be combined with parentheses — evaluate inner parentheses first.
- De Morgan's Laws: ¬(p ∨ q) ≡ ¬p ∧ ¬q and ¬(p ∧ q) ≡ ¬p ∨ ¬q