Choose the Correct or the Most Suitable Answer

This closing 20-question multiple-choice set (Exercise 12.3) draws on every idea across both halves of the chapter. Binary-operation questions (1-7) test the core definition ($S\times S\to S$), which sets fail to be binary under a given operation, and whether a specific $*$ is commutative and/or associative. Logic questions (8-20) move through identifying a true/false statement, counting truth-table rows for a given number of variables, writing the inverse/contrapositive of a conditional, filling in a partially-blank truth table, spotting the one incorrect law among four listed equivalence laws, taking a dual, and classifying a formula as tautology/contradiction/equivalent-to-something-simpler.

Two techniques worth having ready before attempting this set:

• For "inverse of $P\to Q$" and "contrapositive of $P\to Q$" questions, compute $\neg P$ and $\neg Q$ first (applying De Morgan's Laws if $P$ or $Q$ is itself a $\wedge/\vee$ combination), then assemble $\neg P\to\neg Q$ (inverse) or $\neg Q\to\neg P$ (contrapositive) -- never guess the pattern from memory alone. …