Answer to Question #104890 in Discrete Mathematics for Erin

Question

4.2: For the proposition pairs below, create a truth table and compare each proposition’s truth profile: determine whether the pair is logically equivalent, contradictory, consistent or inconsistent.

Example: (¬J ≡ K) with [(J → ¬K) ∧ (¬K → J)]

Write your answer as follows:
Step 1
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T (F) ≡ T (T → F) ∧ (F → T)
T F (F) ≡ F (T → T) ∧ (T → T)
F T (T) ≡ T (F → F) ∧ (F → F)
F F (T) ≡ F (F → T) ∧ (T → F)

Step 2
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T F (F) ∧ (T)
T F T (T) ∧ (T)
F T T (T) ∧ (T)
F F F (T) ∧ (F)

Step 3
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T F F
T F T T
F T T T
F F F F

Answer: (¬J ≡ K) and [(J → ¬K) ∧ (¬K → J)] are logically equivalent


34. (P ∧ Q) ∨ P with (P ∨ Q) ∧ Q


35. [(S → W) → X] with [(S ∧ W) ∨ (W ∧ X)]

Answer 1

34. (P ∧ Q) ∨ P with (P ∨ Q) ∧ Q 

(P ∧ Q) ∨ P (P ∨ Q) ∧ Q

T T (T ∧ T) ∨ T (T ∨ T) ∧ T

T F (T ∧ F) ∨ T (T ∨ F) ∧ F

F T (F ∧ T) ∨ F (F ∨ T) ∧ T

F F (F ∧ F) ∨ F (F ∨ F) ∧ F

(P ∧ Q) ∨ P (P ∨ Q) ∧ Q

T T T ∨ T T ∧ T

T F F ∨ T T ∧ F

F T F ∨ F T ∧ T

F F F ∨ F F ∧ F

(P ∧ Q) ∨ P (P ∨ Q) ∧ Q

T T T T

T F T F

F T F T

F F F F

The statements are not equivalent and are contradictory. The statements are consistent since they are both true for P=Q=T.

35. [(S → W) → X] with [(S ∧ W) ∨ (W ∧ X)]

S W X [(S → W) → X] [(S ∧ W) ∨ (W ∧ X)]

T T T [(T → T) → T] [(T ∧ T) ∨ (T ∧ T)]

T T F [(T → T) → F] [(T ∧ T) ∨ (T ∧ F)]

T F T [(T → F) → T] [(T ∧ F) ∨ (F ∧ T)]

F T T [(F → T) → T] [(F ∧ T) ∨ (T ∧ T)]

T F F [(T → F) → F] [(T ∧ F) ∨ (F ∧ F)]

F T F [(F → T) → F] [(F ∧ T) ∨ (T ∧ F)]

F F T [(F → F) → T] [(F ∧ F) ∨ (F ∧ T)]

F F F [(F → F) → F] [(F ∧ F) ∨ (F ∧ F)]

S W X [(S → W) → X] [(S ∧ W) ∨ (W ∧ X)]

T T T [T → T] [T ∨ T]

T T F [T → F] [T ∨ F]

T F T [F → T] [F ∨ F]

F T T [T → T] [F ∨ T]

T F F [F → F] [F ∨ F]

F T F [T → F] [F ∨ F]

F F T [T → T] [F ∨ F]

F F F [T → F] [F ∨ F]

S W X [(S → W) → X] [(S ∧ W) ∨ (W ∧ X)]

T T T T T

T T F F T

T F T T F

F T T T T

T F F T F

F T F F F

F F T T F

F F F F F

The statements are not equivalent and are contradictory. The statements are consistent since they are both true for S=W=X=T and for S=F, W=X=T.