108年 - 108 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#146879

1. Show that for all sets S and T, $S - T = S \cap \overline{T}$ where $\overline{T}$ is the complement of T. (10 分)

2. Find a minimum cost spanning tree of the following weighted graph. (20 分)

3. A boy needs to eat breakfast and lunch, practice piano, mow the lawn, and read a book toady. In how many ways can he arrange these activities if breakfast must occur before lunch, and at least one other activity must separate the meals? (20 分)

(A) $\sum_{i=0}^m \sum_{j=0}^n (1)$

(B) $\prod_{i=1}^{10} (-1)^i$

(C) $\sum_{i=1}^n i$

(D) $\prod_{i=1}^5 (-i)^{-1}$

(E) $\sum_{i=2}^n \left(1 - \frac{1}{i^2}\right)$

(A) What is the cardinality of the power set $P(A \cup B)$?

(B) What is the set containing all the elements of $A \times B$?

(A) Draw the directed graph representing R.

(B) Determine the transitive closure $R^*$ of R.

(A) $F(x, y, z) = \overline{xy} + xy$

(B) $F(x, y, z) = (x + y + z)(\overline{x} + \overline{y} + \overline{z})$