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首頁 > 研究所、轉學考(插大)◆離散數學 > 112年 - 112 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#144118 > 申論題

題組內容

1. Solve the following recurrence relations. (20%)

(b) Nonhomogeneous recurrence relation: \[a_n = 4a_{n-1} + 8n > 1, n > 0, a_0 = 2;\]

相關申論題

(a) Please show the shortest path. The path should be represented by tracing a sequence of visited vertices, such as (a, b, f, z). (10%)

(b) Please evaluate the length of the shortest path. (10%)

(a) Is relation R antisymmetric? (5%)

(b) Is relation R irreflexive? (5%)

4. Use mathematical induction to prove\[ 2 - 2 \times 9 + 2 \times 9^2 - \cdots + 2 \times (-9)^n = \frac{1 - (-9)^{n+1}}{5} \]whenever ? is a nonnegative integer. (10%)

(a) Please describe the binary relation of partial ordering by using the following example: let T be a set of positive integers, and let R be a binary relation on T such that (x, y) for x, y ∈ T is in R if ? divides y.(5%)

(b) The binary relation can be represented graphically. Represent the elements in Tby points and use arrows to represent the ordered pairs in R. Please show that the above-mentioned directed graph has no cycles if deleting the self-loops from the poset (T, R). (5%)

(c) Please draw Hasse diagrams for the posets (T1,R ) and (T2,R) where\(T_1 = \{8, 168, 4, 112, 28\}\), \(T_2 = \{9, 6, 72, 1512, 108, 3, 36\}\) and \(R\) isdefined above. Please explain whether they are lattices or not. (10%)

(a) Please describe how to calculateb83 mod ? (i.e., the exponent e =(83)10: decimal representation) by the above algorithm and explain it is correct. (10%)

(b) If there is a ?-bit exponent ? with \[ \left\lceil \frac{k}{2} \right\rceil \] bits of “1”, where ⌈ ⌉ denotes aceiling function. How many times of multiplications in this algorithm tocalculate be mod m. Please explain your answer. (10%)

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