Question

a. Show that the following statement forms are all logically equivalent. p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q b. Use the logical equivalences established in part (a) to rewrite the following sentence in two different ways. (Assume that n represents a fixed integer.) If n is prime, then n is odd or n is 2.

Answer 1

To show that the statement forms p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q are logically equivalent, use logical equivalences and truth tables. The sentence 'If n is prime, then n is odd or n is 2' can be rewritten as 'n is prime → (n is odd or n is 2)' or 'n is not prime or (n is odd or n is 2)' using the established logical equivalences.

Step-by-step explanation:

To show that the statement forms p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q are all logically equivalent, we can use logical equivalences and truth tables. Here's how:

1. Start by writing the truth table for p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q.
2. Compare the truth values of the three statement forms in each row of the truth table.
3. If the truth values are the same for all rows, then the statement forms are logically equivalent.

Based on the truth table, we can see that the statement forms p → q ∨ r, p ∧ ∼q → r, and p ∧ ∼r → q are all logically equivalent.

To rewrite the sentence 'If n is prime, then n is odd or n is 2' using the logical equivalences established in part (a), we can use the p → q ∨ r form. We can rewrite it as 'n is prime → (n is odd or n is 2)' or 'n is not prime or (n is odd or n is 2)'.

Answer 2

(a) if n is prime, then n is odd or n is 2

(b) if n is prime and n is not odd, then n is 2

(c) if n is prime and n is not 2, then n is odd

Step-by-step explanation:

a) p → q ∨ r

b) p ∧ ∼q → r

c) p ∧ ∼r → q

Lets show that (a) implies (b) and (c). (a) says that if property p is true, then either q or r is true, thus, if p is true we have:

• If the condition of (b) applies (thus q is not true), we need r to be true because either q or r were true because we are assuming (a) and p. Hence (b) is true
• If the condition of (c) applies (r is not true), since either r or q were true due to what (a) says, then q neccesarily is true, hence (c) is also true.

Now, lets prove that (b) implies (a)

• If p is true and property (b) is true, then if q is true, then either q or r are true thus (a) is correct. If q is not true, then property (b) claims that, since p is true and q not, r has to be true, therefore (a) is valid in this case as well, hence (a) is also true.

(c) implies (a) can be proven with similar argument, changing (b) for (c), q for r and r for q.

With this we prove that the 3 properties are equivalent.

For the rest of the exercise, we have

• property p: n is prime
• property q: n is odd
• property r: n is 2

Translating this, we obtain (a), (b) and (c)

(a) if n is prime, then n is odd or n is 2

(b) if n is prime and n is not odd, then n is 2

(c) if n is prime and n is not 2, then n is odd