Discrete Mathematics — Book-Back Answer Key (Exercise 12.3)
Exercise 12.3 — Book-Back Answer Key
The correct option for each multiple-choice question, with a one-line justification. All twenty textbook answers were independently re-worked and verified; none required correction.
| Q | Answer | Reason |
|---|---|---|
| 1 | (B) \((S \times S) \to S\) | A binary operation maps an ordered pair to a single element of the same set. |
| 2 | (C) \(\mathbb{N}\) | \(2-5=-3\notin\mathbb{N}\), so subtraction is not closed on \(\mathbb{N}\). |
| 3 | (B) Multiplication | The product of two naturals is natural; subtraction and division are not closed. |
| 4 | (D) \(a * b = a^{b}\) | \((-2)^{1/2}\notin\mathbb{R}\), so \(a^{b}\) can leave \(\mathbb{R}\). |
| 5 | (B) \(\mathbb{Z}\) | \(1*1=\tfrac17\notin\mathbb{Z}\); the other sets are closed under \(\tfrac{ab}{7}\). |
| 6 | (B) \(y = -\dfrac{2}{3}\) | \(3\odot(y\odot5)=24y+23=7\Rightarrow y=-\tfrac23\). |
| 7 | (C) both commutative and associative | \(\sqrt{a^{2}+b^{2}}\) is symmetric, and both groupings give \(\sqrt{a^{2}+b^{2}+c^{2}}\). |
| 8 | (D) \(\sqrt{5}\) is an irrational number | Only '\(\sqrt5\) is irrational' is true; the other three are false. |
| 9 | (C) Chennai is in China or \(\sqrt{2}\) is an integer | Only this disjunction has both parts false: \(F\lor F=F\). |
| 10 | (B) \(8\) | \(2^{3}=8\) rows. |
| 11 | (D) \((\lnot p \land \lnot q) \to (\lnot p \lor \lnot q)\) | Inverse negates both sides: \((\lnot p\land\lnot q)\to(\lnot p\lor\lnot q)\) by De Morgan. |
| 12 | (A) \(\lnot r \to (\lnot p \land \lnot q)\) | Contrapositive: \(\lnot r\to\lnot(p\lor q)=\lnot r\to(\lnot p\land\lnot q)\). |
| 13 | (C) \(F,\ T,\ T,\ T\) | Column \(F,T,T,T\): \((p\land q)\to\lnot q\) is false only in the \(TT\) row. |
| 14 | (C) \(3\) | \(\lnot(p\lor\lnot q)\) is true once and false three times. |
| 15 | (C) \(\lnot(p \lor q) \equiv \lnot p \lor \lnot q\) | De Morgan needs \(\land\): \(\lnot(p\lor q)\equiv\lnot p\land\lnot q\), not \(\lnot p\lor\lnot q\). |
| 16 | (B) \(F,\ T,\ T,\ T\) | Column \(F,T,T,T\): \((p\land q)\to\lnot p\) is false only in the \(TT\) row. |
| 17 | (D) \(\lnot(p \land q) \land [\,p \land (p \lor \lnot r)\,]\) | Swap every \(\lor\) and \(\land\): \(\lnot(p\land q)\land[p\land(p\lor\lnot r)]\). |
| 18 | (C) logically equivalent to \(p \land q\) | Distribute: \((p\land\lnot p)\lor(p\land q)\equiv p\land q\). |
| 19 | (A) \(F,\ T,\ F,\ T\) | \(F,T,F,T\): each 'and' is true only when both arithmetic parts hold. |
| 20 | (D) If \(p\) and \(q\) are any two statements, then \(p \leftrightarrow q\) is a tautology | \(p\leftrightarrow q\) is a contingency, not a tautology. |