Homework 4
Question 1:
A = there have been three previous female Presidents of the United States F
B = peanuts contain saturated fat T
C = Beyonc ́e performed at the Superbowl 50 halftime show T
D = Luke Skywalker will be unveiled as the father of Rey in Episode VIII U
- A ∨ B = T
- A ∧ B = F
- A ⇒ B = T
- A ⊕ B = T
- B ∨ A = T
- B ∧ A = F
- B ⇒ A = F
- B ⊕ A = T
- C ∨ D = T
- (¬C) ∧ (¬D) = F ∧ U = F
- A ⇒ D = F ⇒ U = F
- A ⇒ A = T
- D ⇒ D = (T ⇒ T) or (F ⇒ F) = T
- ¬((¬A) ∧ (¬B)) = ¬(T ∧ F) = ¬(F) = T
- ¬((¬B) ∧ (¬C)) = ¬(F ∧ F) = ¬(F) = T
- ¬((¬A) ∧ (¬B) ∧ (¬C)) = ¬(T ∧ F ∧ F) = ¬(F) = T
- D ∧ ¬D = (T ∧ F) or (F ∧ T) = F
- D ∨ ¬D = (T ∨ F) or (F ∨ T) = T
Question 2:
((¬G) ⊕ H) ⇔ ((G ∨ ¬I) ∧ F )
| F | G | H | I | ¬G | (¬G) ⊕ H | ¬I | G ∨ ¬I | (G ∨ ¬I) ∧ F | ((¬G) ⊕ H) ⇔ ((G ∨ ¬I) ∧ F ) |
| T | T | T | T | F | T | F | T | T | T |
| T | T | T | F | F | T | T | T | T | T |
| T | T | F | T | F | F | F | T | T | F |
| T | T | F | F | F | F | T | T | T | F |
| T | F | T | T | T | F | F | F | F | T |
| T | F | T | F | T | F | T | T | T | F |
| T | F | F | T | T | T | F | F | F | F |
| T | F | F | F | T | F | T | T | T | F |
| F | T | T | T | F | T | F | T | F | F |
| F | T | T | F | F | T | T | T | F | F |
| F | T | F | T | F | F | F | T | F | T |
| F | T | F | F | F | F | T | T | F | T |
| F | F | T | T | T | F | F | F | F | T |
| F | F | T | F | T | F | T | T | F | T |
| F | F | F | T | T | T | F | F | F | F |
| F | F | F | F | T | F | T | T | F | T |
Question 3:
Celebrity(x) — the proposition that x is well-known in pop American culture
Wedded(x,y) — the proposition that x and y at one point in their lives got married
Married(x,y) — the proposition that x and y are currently married
Single(x) — the proposition that x is not currently married
Divorced(x,y) — the proposition that x and y at one point in their lives got divorced
- Donald Knuth is not a celebrity.
¬Celebrity(Donald Knuth)
- Kim Kardashian is a celebrity.
Celebrity(Kim Kardashian)
- Kim Kardashian is married to a celebrity.
∃ x Celebrity(x) ∧ Married(Kim Kardashian, x)
- Kim Kardashian wedded two different celebrities.
∃ x ∃ y Celebrity(x) ∧ Celebrity(y) ∧ Wedded(Kim Kardashian, x) ∧ Wedded(Kim Kardashian, y)
- Some celebrities marry other celebrities, while some just marry ordinary folk.
∃ x ∃ y ∃ z ∃ u Celebrity(x) ∧ Celebrity(y) ∧ Celebrity(z) ∧ ¬Celebrity(u) ∧ Wedded(x,y) ∧ Wedded(z,u)
- If you’re married, then you’re not single.
∀ x ∀ y Married(x,y) ⇒ ¬Single(x)
- It’s possible to have once been married, and never divorced, and yet be currently single.
¬(∀ x ∀ y Wedded(x,y) ∧ ¬Divorced(x,y) ⇒ ¬Single(x))
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