Question

1. Simplify the statement (? ∧ (~(~? ∨ ?))) ∨ (? ∧ ?). Show all steps in the simplification process.

2. Let ? be some statement based on the simple statements ?, ?, and ?. Write and simplify the form of ? based on the following truth table for ?:

ST T F F F T T F rT F T F T F T F q-T T F F T T F F

3. There are four treasure chests deep inside of a cave. Each treasure chest has a sign underneath it.

Treasure Chest #1: The treasure is not here.

Treasure Chest #2: Chest #3 is empty.

Treasure Chest #3: The treasure is not in Chest #4.

Treasure Chest #4: Chest #2 tells the truth.

A big sign on the cave wall reads, “At least two of these treasure chests are lying.”

Where is the treasure?

4. The conjunction ? ∧ ? can be written with Sheffer strokes as (?|?)|(?|?). Use Sheffer strokes to write ? ∧ (~? ∨ ?).

5. One of DeMorgan’s Laws tells us that ~(? ∨ ?) ≡ ~? ∧ ~?. There is a third equivalence that can be written using the Pierce arrow: ? ↓ ?. Using this fact, plus other facts of logical equivalence, write the conditional statement ? → ? using only Pierce arrows.

Answer 1

(1)

Given statement is [p ^ {~(~p v q)}] v (p ^ q)

= [p ^ (p ^ ~q)] v (p ^ q)

= (p ^ ~q) v (p ^ q)