1. Use natural deduction to prove that the following arguments are valid
a) P ∧ Q ∴ P ∨ Q
1. (P ∧ Q) → (P ∨ Q)
2. (P ∨ Q) ∧ (P ∨ R) → (Q ∨ R)
3. (P ∨ Q) ∧ (P ∨ R) ∨ (Q ∨ R) by implication law
4. (P ∨ Q) ∨ (P ∨ R) ∨ (Q ∨ R) by de Morgan’s law
5. [(P ∧ Q) ∨ (P ∧ R)] ∨ (Q ∨ R) by de Morgan’s law
6. (P ∧ Q) ∨ Q] ∨ [(P ∧ R) ∨ R] by commutative and associative laws
7. [(P ∨ Q) ∧ (Q ∨ Q)] ∨ [(P ∨ R) ∧ (R ∨ R)] by distributive laws
8. (P ∨ P) ∨ (Q ∨ R) by communicative and associative laws
b) (P ∧ Q) → R, P, Q ∴ R
2. (P → Q) ∨ (P → R)
3. (¬P ∨ Q) ∨ (¬P ∨ R) by implication law, twice
4. (¬P ∨ ¬P) ∨ (Q ∨ R) by commutative and associative laws
5. ¬P ∨ (Q ∨ R) by Idempotent laws
6. p → (Q ∨ R) by implication law
7. P → Q, P ∨ Q ∴ Q
8. Q → R
2. Formalise the following statements using propositional logic. Remember to provide a key that shows what your atomic sentence letters stand for. (5 marks each, 30 marks total)
a) If you don’t wear a coat you’ll get wet.
¬ C → W
b) I’ll be back on Monday or Tuesday.
B → (M ∨ T)
c) You will only be admitted if you have an invitation.
A → I
d) He’s young, not stupid.
Y → ¬S
e) I won’t be at work on Thursday or Friday.
¬ W → (T ∨ F)
f) You will be admitted if and only if you have an invitation and arrive on time.
A ↔ (I ∨ T)
3. Translate the following expressions of propositional logic into English. (5 marks each, 20 marks total) Key:
A: Apples are fruits
B: Bananas are fruits
C: Cucumbers are fruits
L: Lettuces are fruits
T: Tomatoes are fruits
a) A ∧ B
Apples and bananas are fruits.
b) ¬(L ∨ C)
Neither lettuce nor cucumbers are fruits.
c) L → (C ∧ T)
If lettuces are fruits, then cucumber and tomatoes are fruits.
d) ¬A ↔ T
Both apples and tomatoes are not similar.
4. Symbolise the following arguments and then use natural deduction to prove that they are valid. Remember to provide a symbolisation key. (10 marks each, 20 marks total)
KEY:
R Rain
P pour
U umbrella
W wet
a) It either rains or it pours. If it pours, then it rains. So it rains.
(R ∨ P) → (P ∨ R) ∴ R
1. (R ∧ P) → (P ∨ R)
2. (P ∨ Q R) ∧ (P ∨ R) ∨ (P ∨ R) by implication law
3. (P ∨ R) ∨ (P ∨ R), (R) by de Morgan’s law
4. P → R
b) If it rains and I don’t have an umbrella, I’ll get wet. If it rains and I have an umbrella, a car will splash me and I’ll still get wet. So if it rains I’ll get wet.
1. R U W → R, U, S, W ∴ R ∧ W
2. R → (U, W) (U, S)
3. R → U (W,S)
