Week 2 - Practice Problems
Symbolic Form Examples
Translate the following statements in symbolic form:
-
It is not hot (H), but it is sunny (S).
Answer: ¬H ∧ S -
It is neither hot (H) nor sunny (S).
Answer: ¬H ∧ ¬S -
2 is a positive integer (P) or √2 is a rational number (R).
Answer: P ∨ R -
The crop (C) will be destroyed if there is a flood (F).
Answer: F ⟶ C -
Let P: You drive over 80kms per hour, Q: You get a speeding ticket. If you do not drive over 80kms per hour, then you will not get a speeding ticket.
Answer: ¬P ⟶ ¬Q -
Let P: You drive over 80kms per hour, Q: You get a speeding ticket. Driving over 80kms per hour is sufficient for getting a speeding ticket.
Answer: P ⟶ Q -
He swims (S) if and only if the water is warm (W).
Answer: S ⟷ W
Negation Examples
-
Negate P: Varun’s smartphone has at least 32 GB of memory (M).
Answer: Varun’s smartphone has less than 32 GB of memory (¬M). -
P: Roses are red (R). Negate P.
Answer: Roses are not red (¬R). -
Negate “2 + 4 = 6 (A) and 7 < 12 (B)”.
Answer: (¬A) or (¬B) -
Negate “9 is greater than 4 (G) or 6 is less than 8 (L).”
Answer: (¬G) and (¬L) -
Negate “If he studies (S), he will pass the examination (P).”
Answer: S and (¬P) -
Negate “He swims (S) if and only if the water is warm (W).”
Answer: (S and ¬W) or (¬S and W)
Truth Table Examples
Example 1: P ⟶ Q ≡ ¬P ∨ Q
| P | Q | ¬P | ¬P ∨ Q |
|---|---|---|---|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
Example 2: P ⟷ Q ≡ (P ⟶ Q) ∧ (Q ⟶ P)
| P | Q | P ⟷ Q | P ⟶ Q | Q ⟶ P | (P ⟶ Q) ∧ (Q ⟶ P) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | F | T | F |
| F | T | F | T | F | F |
| F | F | T | T | T | T |
Example 3: ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
| P | Q | ¬P | ¬Q | P ∨ Q | ¬(P ∨ Q) | ¬P ∧ ¬Q |
|---|---|---|---|---|---|---|
| T | T | F | F | T | F | F |
| T | F | F | T | T | F | F |
| F | T | T | F | T | F | F |
| F | F | T | T | F | T | T |
- Example 4: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
| P | Q | ¬P | ¬Q | P ∧ Q | ¬(P ∧ Q) | ¬P ∨ ¬Q |
|---|---|---|---|---|---|---|
| T | T | F | F | T | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | F | T | T | F | T | T |
- Example 5: P ∨ (P ∧ Q) ≡ P ∧ (P ∨ Q) ≡ P
| P | Q | P ∧ Q | P ∨ (P ∧ Q) | P ∨ Q | P ∧ (P ∨ Q) |
|---|---|---|---|---|---|
| T | T | T | T | T | T |
| T | F | F | T | T | T |
| F | T | F | F | T | F |
| F | F | F | F | F | F |
- Example 6: Tautology
| P | ¬P | P ∨ ¬P |
|---|---|---|
| T | F | T |
| F | T | T |
- Example 7: Contradiction
| P | ¬P | P ∧ ¬P |
|---|---|---|
| T | F | F |
| F | T | F |
