Question

Shankar and Jwala are both members of a Youth club, though they are not speaking to each other and refuse to work with each other. Chaya, the club president, is appointing members to the fundraising committee, but she has resolved that she will not appoint anyone without his or her explicit consent. Shankar says, "I will not consent unless I know whether Jwala is to be a member." Jwala says, "I will not consent unless I know whether Shankar is to be a member." If all three stick to these resolutions, then:

• A. Neither of them can be appointed to the committee.
• B. They must either both be appointed or both left out.
• C. The committee may have one of them, both of them, or neither.
• D. Either one can be appointed, but not both.

Hint:

Check whether both being appointed together creates a circular problem where each person's consent depends on the other's status being decided first, then check what happens if only one of them is appointed.

Answer 1

The Correct Option is D

Step 1: Understand what each person's condition means.
Shankar will only give his consent once he knows whether Jwala is going to be on the committee. Jwala will only give her consent once she knows whether Shankar is going to be on the committee. Chaya will not appoint either of them without that consent.

Step 2: Test the case where both are appointed.
For Shankar to consent, he needs to already know Jwala's status, meaning Jwala's spot must be decided before Shankar agrees. But Jwala's spot depends on her own consent, which in turn needs her to already know Shankar's status. Each one's decision is waiting on the other one's decision first, with no starting point. This loop can never resolve, so both being appointed together is not possible.

Step 3: Test the case where only Shankar is appointed.
If Jwala is left out, then Shankar can be told plainly that Jwala is not going to be a member. Shankar now knows Jwala's status, so his condition is satisfied and he can give consent. Nothing stops Jwala's status from being settled first in this case, since she is simply not chosen.

Step 4: Test the case where only Jwala is appointed.
By the same reasoning, if Shankar is left out, Jwala can be told he is not a member, her condition is met, and she can consent.

Step 5: Test the case where neither is appointed.
This case is also consistent on its own, but the question is asking what must happen given the resolutions, not just what is possible. The two working, non-circular cases are exactly the ones where one of them is in and the other is out.

Final Answer:
The circular dependency rules out having both Shankar and Jwala on the committee together, while placing just one of them works fine either way. So either one can be appointed, but never both at once. \[ \boxed{\text{Either one can be appointed, but not both.}} \]