102年 - 102 國立中山大學_碩士班招生考試_電機系(丙組)：離散數學#108985

(a) Fundamental Theorem of Arithmetic

(b) Homeomorphic graph

(c) Equivalence relation

(d) Four-color Theorem

(a) How many positive divisors does x have? [5%]

(6) How many positive divisors of x that are divisible by 252? [5%]

(c) Determine how many positive divisors of x are perfect squares? [5%]

(a) If an equivalence relation R on set A = {1,2,3,4,5}induces the partition A = {1,3} ∪{2,4} ∪{5}, what is R? [5%]

(b) Let R={(1,1),(1,2),(2,2),(2,4), (3,3),(3,4),(4,5),(5,5)}be a relation on A. What is the relation R³ ? [5%]

4.What is the Ferrers graph? Use it to explain the statement "The number of partitions of an integer n into m summands is equal to the number of partitions of n into summands where m is the largest summand". [10%]

5. Use the generating function to find the number of integer solutions of the equationx₁ +x₂ +x₃ +x₄ =20, where -3 ≤ x₁,-3 ≤ x₂,-5 ≤ x₃ ≤5, and 0≤x₄.[15%]

6. Find the number of colors needed to color the following map so that no two adjacent regions have the same color. [15%]

7. d the number of permutation of the letters x, x, y, y, z, z so that no x appears in the firs positions, no y appears in the third position and no z appears in the fifth and sixth positions. [15%]

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102年 - 102 國立中山大學_碩士班招生考試_電機系(丙組)：離散數學#108985

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