Answer to Question #111220 in Discrete Mathematics for Martin

Question

f) Suppose we are told that the following expression is true: P ↔ (Q ∧ ¬Q). What can we then
conclude about P's truth?

1. P must be true
2. It is uncertain whether P is true or false
3. P must be false.


g) Which of the following informal arguments is the best example of a use of conditional elimination?
In addition to being able to recognize conditional elimination you will need to use your knowledge
of the correct translation of natural language conditionals.

1. “John will prove a theorem only if he isn't very tired. He slept very well last night, so he'll
prove a theorem."
2. “John won't prove theorems if he is tired. He's pretty tired today, so he'll prove a theorem."
3. ”If Dave doesn't sleep, then he'll prove a theorem. Since he slept soundly, he
consequently proved no theorems."
4. “Dave has a good night's sleep only if he will prove a theorem. Last night, he slept quite
well and so he will prove a theorem.

Answer 1

f) P ↔ (Q ∧ ¬Q) is True.

$P ↔ (Q ∧ ¬Q) \equiv P ↔ F$ ; where F denotes False.

$\implies P ↔ (Q ∧ ¬Q) \equiv T \equiv P ↔ F$

Thus, P must be False as well.

Option 3 is the correct answer.

g) The option 1 is correct as it translates to $P \to Q; P \implies Q$ This simply a rule of inference known as Modus Ponens.

Option 1 is the correct answer.