Question

Let P, Q, and R be the compound propositions P:(p19) (p Vr), Q : (p) ( pr), R:(qer).

1. Make a truth table with the last three columns labeled by P, Q, and R.
2. Which of the six possible logical implications (see below) between pairs of P, Q, and R are true, and why? Are any of P, Q, and R logically equivalent to another?

Answer 1

A truth table is created to find the values of P, Q, and R for all possible combinations of p and r. Logical implications between pairs of P, Q, and R are determined by comparing their truth values. P implies Q and R implies Q, but Q does not imply P or R. None of P, Q, and R are logically equivalent to each other.

Step-by-step explanation:

Let's create a truth table to find the values of P, Q, and R:

P: (p∨ r)Q

: (p) ∧ (p ∨ r)

R: (q ∧ r)
p
r
P
Q
R
T
T
T
T
F
T
F
T
T
F
T
T
T
T
F
F
F
F
F

The truth table shows the values of P, Q, and R for all possible combinations of p and r.

To determine the logical implications between pairs of P, Q, and R, we can compare the truth values. A logical implication is true when the truth value of the antecedent (the proposition on the left-hand side) implies the truth value of the consequent (the proposition on the right-hand side).

In this case, we can see that P implies Q, because whenever P is true, Q is also true. However, Q does not imply P, as there are some cases where Q is true but P is false. Similarly, R implies Q, but Q does not imply R. None of P, Q, and R are logically equivalent to each other, as their truth values do not match for all possible combinations of p and r.