1.Suppose we are given logic statements p, q, and r.
a. If p → q and p → r, may we conclude that q → r? Answer yes or no and give a reason why.
b. What is the converse of the following: not p → not q
c. What is the contrapositive of r → q?
d. The statements p → q and q → r are given. If we know q is true, is p necessarily true? Explain.
(a) No
If it rains, I'll eat chicken.
If it rains, I'll go outside.
No way to conclude that eating chicken means going outside.
(b) Come on. You know that the converse of p→q id q→p. So, ...
(c) ~q → ~r
(d) Nope. I'm sure you can come up with an example. (Use what you know about converses.)
a. No, we cannot conclude that q → r.
To determine whether we can deduce q → r, we need to analyze the given statements. In this case, we have p → q and p → r. The condition for being able to infer q → r from these statements is that if the antecedent (q) of one implication (p → q) is true, then the consequent (r) of the other implication (p → r) must also be true.
However, in this scenario, we do not have any direct relationship between q and r. Therefore, we cannot conclude that q → r based solely on the given statements p → q and p → r.
b. The converse of "not p → not q" is "not q → not p."
Converse of a logical statement involves interchanging the positions of the antecedent and the consequent. In this case, the original statement is "not p → not q." To form its converse, we swap the positions of "not p" and "not q" to obtain "not q → not p."
c. The contrapositive of the statement "r → q" is "not q → not r."
To construct the contrapositive of a logical statement, we need to negate both the antecedent and the consequent and then flip their positions. In this case, the original statement is "r → q." Therefore, the contrapositive would be "not q → not r."
d. No, the truth value of p cannot be determined solely based on the given information.
Given the statements p → q and q → r, if we know that q is true, it does not necessarily mean that p is also true. The implication p → q only establishes a relationship between p and q, indicating that if p is true, then q must be true, but it does not provide any information about the truth value of p in the absence of q. Therefore, without further information, we cannot conclude the truth value of p when we know q is true.