1. Steve Losh
  2. introduction-to-mathematical-thinking

Source

introduction-to-mathematical-thinking / week-02 / assignment-03.markdown

Assignment 2

Problems

Problem 1

  1. T ⇒ (D ∧ Y)
  2. T ⇒ (Y ⇒ ¬D)
  3. T ∧ (T ⇒ ((D ∧ Y) ∨ (¬D ∧ ¬Y)))

Problem 2

φ   ¬φ   ψ   (φ ⇒ ψ)   (¬φ ∨ ψ)
-------------------------------
T    F   T   T         T
T    F   F   F         F
F    T   T   T         T
F    T   F   T         T

Problem 3

That φ ⇒ ψ and ¬φ ∨ ψ are logically equivalent.

This makes sense, because the implication is always true if φ is false. Otherwise it's only true when ψ is true. Which is exactly what ¬φ ∨ ψ says.

Problem 4

φ   ψ   ¬ψ    (φ ⇒ ψ)   (φ ⇏ ψ)   (φ ∧ ¬ψ)
------------------------------------------
T   T   F     T         F         F
T   F   T     F         T         T
F   T   F     T         F         F
F   F   T     T         F         F

Problem 5

That (φ ⇏ ψ) and (φ ∧ ¬ψ) are logically equivalent.

This makes sense, because the only way for φ to "not imply" ψ is for φ to be true but ψ be false, which is exactly what (φ ∧ ¬ψ) says.