Final answer:
Only the proposition c. p→q implies (¬p)∨q. Propositions a, b, d, and e do not imply (¬p)∨q because they do not satisfy the condition for all possible truth values of p and q.
Step-by-step explanation:
The question asks which of the propositions imply (¬p)∨q. Let's evaluate each option:
- a. ¬q: This proposition does not imply (¬p)∨q since (¬q) by itself does not tell us anything about p or its relationship to q.
- b. p∨q: This proposition directly includes q, but it does not imply (¬p)∨q by itself, since p could be true, and hence (¬p) would be false.
- c. p→q: This is the material implication, which means if p is true, then q must also be true. If p is false, then (¬p) is true, making the entire proposition (¬p)∨q true regardless. So, p→q does imply (¬p)∨q.
- d. p∧q: This proposition suggests both p and q are true simultaneously, but it does not imply (¬p)∨q because it does not address the case where p could be false.
- e. p⊕q (exclusive or): The 'exclusive or' means that either p or q is true, but not both. This does not necessarily imply (¬p)∨q, since if p is true and q is false, (¬p) would be false, and hence (¬p)∨q would be false as well.
Therefore, out of the given propositions, only the conditional p→q implies (¬p)∨q.