111年 - 111 國立嘉義大學_碩士班招生考試試題_資訊工程學系：離散數學#145438

1. Simplify the expression \[ \overline{\overline{(A \cup B)} \cap C \cup \overline{B}} \]. (10%)

2.Determine the number of nonnegative integer solutions to the equation \( x_1 + x_2 + x_3 + x_4 = 18 \) and \( x_i \le 7 \) all \( i \). (20%)

3. In how many ways can a police captain distribute 24 rifle shells to four police officers so that each officer gets at least three shells, but not more than eight? (20%)

(a) \( P(\{a, b\})) \), where \( P(A) \) represents the power of set of Set \( A \). (5%)

(b)\( \{\{\emptyset\}\} \oplus \{a, \emptyset, \{\emptyset\}\} \cdot (5\%) \)

(c) {a, ∅, {∅}} − {∅} . (5%)

(d) {∅} ∩ P({∅}) . (5%)

(a) \( f(x) = 10 - x \) . (5%)

(b) \( f(x) = \lfloor x/2 \rfloor \). (5%)

(c) \( f(x) = x - \left\lfloor \frac{x}{2} \right\rfloor \). (5%)

(d) \( f(x) = x^2 + 5 \) . (5%)

6. Given two propositions p and q, please show that\[ \overline{p \lor q} \leftrightarrow \bar{p} \land \bar{q} \]is a tautology. (10%)