Final answer:
Among the provided statements, A and D are true, while B and C are false. Statement A is true because the disjunction of two true statements is true. Statement D is true because if a conjunction and one of its parts are true, the other part must also be true.
Step-by-step explanation:
The question is concerning the validity of logical statements and their relationships. Let's analyze each option:
- A) If p, q, and r are true, then q + r is true. This statement is true because if q and r are individually true, their disjunction (represented symbolically as q + r) would also be true.
- B) If p + q and q r are true, then p is true. This statement is false because p + q being true does not guarantee that p is true since the true value could come from q. Therefore, without additional information about q, we cannot conclude that p is true.
- C) If p q and q + r are true, then r is true. This statement is false. Although q + r is true, it does not necessarily mean that r is true because q could be the true statement making the disjunction true.
- D) If p q and p are true, then q is true. This statement is true. If p q (conjunction) is true and p is also true, then q must be true for the conjunction to hold.
Other examples in the provided information that demonstrate the property of truth are: the commutative property (A + B = B + A), which holds true for addition in mathematics, and the fact about independent events in probability (P(A AND B) = P(A)P(B) is true for independent events).