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COS2661 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS }
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COS2661 EXAM PACK 2025 {DETAILED QUESTIONS AND ANSWERS }
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QUESTION 1 [15]
For the following sentence determine whether the indicated pairs of sentences are tautologically
equivalent. Use truth tables to show your answer.
1.1 A ↔ B and (A ù B) ú (¬A ù ¬B) (4)
A B A↔B (A ù B) ú (¬A ù ¬B)
T T T T
T F F F
F T F F
F F T T
A ↔ B is tautologically equivalent to (A ù B) ú (¬A ù ¬B)
1.2 (A ù B) → C and A → (B → C) (4)
A B C (A ù B) → C A → (B → C)
T T T T T
T T F F F
T F T T T
T F F T T
F T T T T
F T F T T
F F T T T
F F F T T
(A ù B) → C is tautologically equivalent to A → (B → C)
1.3 A ↔ (B ↔ (C ↔ D)) and ((A ↔ B) ↔ C) ↔ D (7)
A B C D A ↔ (B ↔ (C ↔ D)) ((A ↔ B) ↔ C) ↔ D
T T T T T T
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T T T F F F
T T F T F F
T T F F T T
T F T T F F
T F T F T T
T F F T T T
T F F F F F
F T T T F F
F T T F T T
F T F T T T
F T F F F F
F F T T T T
F F T F F F
F F F T F F
F F F F T T
A ↔ (B ↔ (C ↔ D)) is tautologically equivalent to ((A ↔ B) ↔ C) ↔ D
QUESTION 2
[10]
Study the table below and, translate the following English sentences into FOL.
English FOL
Names Max, Claire, Jenny, Nancy Same
Predicates x marry y x Marry (x, y) x
is y =y
x is in love with y x InLove (x, y)
is intelligent. Intelligent (x)
Functions the oldest daughter of x the o-daughter (x) y-daughter
youngest daughter of x (x)
2.1 Max can marry either Nancy's oldest daughter or her youngest daughter (in a monogamous
society).
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2.2 Jenny is Nancy's youngest daughter and Claire is her oldest daughter.
2.3 Neither Claire nor Jenny is in love with Max.
2.4 Jenny will not marry Max unless he is intelligent and in love with her.
2.5 Max is not both intelligent and in love with Jenny.
(1) [Marry (Max, o-daughter(Nancy)) ú Marry(Max, y-daughter (Nancy))] ù ¬[Marry (Max,
odaughter(Nancy)) ù Marry (Max, y-daughter (Nancy))] exclusive sense of OR!
(2) Jenny = y-daughter (Nancy) ù Claire = y-daughter (Nancy)
(3) ¬ InLove(Claire, Max) ù ¬ InLove(Jenny, Max) OR
¬[ InLove(Claire, Max) ú InLove(Jenny, Max)] complete denial!
(4) ¬Marry (Jenny, Max) ú (Intelligent (Max) ù InLove(Max, Jenny)) the easiest! OR
¬(Intelligent (Max) ù InLove(Max, Jenny)) → ¬Marry (Jenny, Max) closer to common
sense!
OR
Marry (Jenny, Max) → (Intelligent (Max) ù InLove(Max, Jenny)) closer to logical
sense!
(5) ¬ (Intelligent(Max) ù InLove(Max, Jenny)) OR
¬Intelligent(Max) ú ¬InLove(Max, Jenny) partial denial!
QUESTION 3
[10]
Translate the following formal language of Tarski's World (FOL) sentences into English sentences.
3.1 (FrontOf(a, c) ù FrontOf(b, c)) ù (Smaller(a, c) ù Smaller(b, c))
3.2 ¬Between (c, a, b) ù ¬ (FrontOf(c, a) ú FrontOf(c, b))
3.3 ¬(FrontOf(d, b) ú FrontOf(d, e)) ù ¬(FrontOf(c, b) ú FrontOf(c, e))
3.4 (Between (a, c, d) ú Between (b, c, d)) ù ¬ (Between (a, c, d) ù Between (b, c, d))
3.5 [(Small(a) ù Cube(a) ù Large(b) ù Dodec(b) ù FrontOf(a, b)] ù [LeftOf(a, b) ú (Tet(c) ù Tet(d) ù
Between(a , c, d))]
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