Chapter 1 - Section 1.2 - Applications of Propositional Logic - Exercises - Page 24: 34

It is possible, Vijay and Kevin are chatting, Randy, Abby and Heather are not.

Lets list the statements 1) Either Kevin or Heather, or both, are chatting. 2) Either Randy or Vijay, but not both, are chatting. 3) If Abby is chatting, so is Randy. 4) Vijay and Kevin are either both chatting or neither is. 5) If Heather is chatting, then so are Abby and Kevin. Rewrite the statements using propositional logic notation. The first letter of the name of one of the friend means that person is chatting. E.g. $K$ means"Kevin is chatting". All the statements below have to be true and our goal is to find whether $K, H, R, V, A$ are true or false. 1) $K\lor H$ 2) $R \oplus V$ 3) $A \to R$ 4) $V \leftrightarrow K$ 5) $H \leftrightarrow (A \land K)$ Lets start from statement 2. There are 2 possibilities for statement 2 to be true; $R$ true, $V$ false and $V$ true, $R$ false. We will examine both of those possibilities. First lets assume $R$ is true. Then $V$ is false. For statement 4 to be true $K$ is also false. For statement 1 to be true $H$ is true. But now, since $H$ is true, from statement 5 $A \land K$ has to be true, so $A$ and $K$ both have to be true. But we already know $K$ is not true. We have arrived to a contradiction. There is no solution where $R$ would be true and $V$ false. So, $R$ is true, $V$ is false is not possible. Lets try the other option. $V$ is true, $R$ is false For statement 3 to be true, if $R$ is false, $A$ has to be false. For statement 4 to be true, if $V$ is true, $K$ also has to be true. If $A$ is false, $A \land K$ is also false. Now, for statement 5 to be true, $H$ has to be false. So, $V, K$ are true, and $R, A, H$ have to be false.