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: Set up two yes/no facts to track.
Let $S$ = "Shankar is appointed" and $J$ = "Jwala is appointed". There are four possible combinations of $S$ and $J$: both true, both false, or exactly one true.

Step 2: Check the case $S$ = true and $J$ = true.
Shankar's consent needs $J$'s value to be already known. Jwala's consent needs $S$'s value to be already known. Since both are being decided together, neither value can be "already known" before the other is settled, so this case can never actually be reached through the resolutions given. It is logically blocked.

Step 3: Check the case $S$ = true and $J$ = false.
Jwala is simply not chosen, so her own consent is never needed. Shankar is told $J$ is false, satisfying his condition, so he consents, and Chaya can appoint him. This case works cleanly.

Step 4: Check the case $S$ = false and $J$ = true.
By the same logic with the roles swapped, this case also works cleanly: Shankar is left out, Jwala is told $S$ is false, she consents, and she is appointed.

Step 5: Check the case $S$ = false and $J$ = false.
Neither needs to consent to anything, since neither is being appointed, so there is no contradiction, but this is a trivial case, not a forced outcome.

Step 6: Combine the results.
Out of the two non-trivial live scenarios, exactly one of $S, J$ is true and the other is false. The both-true case is impossible, so the two of them can never serve together, only separately.

Final Answer:
Either Shankar or Jwala can be appointed on his or her own, but the two of them can never both be on the committee at the same time. \[ \boxed{\text{Either one, but never both}} \]