Proposition on scenarios

Question

There are 3 statements with the following meanings:

A: Annie came first in sports
B: Jane came first in sports
C: Erick came first in sports

Use A,B,C to write proposition that is True if and only if the first winner is one and only one of Annie, Jane and Erick. The proposition must be in conjunctive normal form.

The proposition i came up with is:

(B∨C) ∧ (A∨C) ∧ (A∨B)

I'm having some issue regarding the statement mentioning "one and only" which i don't quite fully understand. Does the proposition i came out with satisfy the condition?

Answer 1

$A\land - B\land - C $ is true iff Annie wins and Jane and Erick lose. This should help. ..

Answer 2

Your sentence is saying that at least one of Jane and Erick, and that at least one of Annie and Erick, and at least one of Annie and Jane came in first. This means that at least two of them would have to come in first, and possibly all three.

OK, so that is not what you want.

Instead, you can list all the possible scenarios where exactly one of them came in first, and then say that at least one of those scenarios holds.

So, we could say:

Scenario 1: Annie came in first, and Jane and Erick did not. This translates to $A \land \neg B \land \neg C$

Scenario 2: Jane came in first and Annie and Erick did not.

Scenario 3: ...

And now do:

Scenario1 $\lor$ Scenario2 $\lor$ Scenario3

Finally, note that these scenarios are mutually exclusive (e.g in Scenario 1 we have $A$ but in Scenario 2 we have $\neg A$ and so exactly one of these scnearios will hold, as desired.