Answer
(a)
The given compound statement can be represented in simple statements as,
\[\begin{align}
& p:\text{ You do homework right after class}. \\
& q:\text{ You will fall behind}\text{.} \\
\end{align}\]
Use the representation to re-write the provided statement as,
If p, then not q, and if not p, then q.
‘If-then’ is represented by the symbol ‘\[\to \]’, ‘Not or negation’ is represented by the symbol ‘\[\sim \]’, and ‘And’ is represented by the symbol ‘\[\wedge \]’.
Use the symbols to determine the symbolic form of the provided statement as,
\[\left( p\to \sim q \right)\wedge \left( \sim p\to q \right)\]
Hence, the symbolic form for the provided statement is, \[\left( p\to \sim q \right)\wedge \left( \sim p\to q \right)\].
(b)
The truth table is as follows:
(c)
From the truth table, it is clear that \[\left( p\to \sim q \right)\wedge \left( \sim p\to q \right)\] is true only when \[\left( p\to \sim q \right)\] and \[\left( \sim p\to q \right)\] are true.
Hence, the provided statement is false when\[\left( p\to \sim q \right)\] and \[\left( \sim p\to q \right)\]have different truth values.
