离散数学英文试题A

西南大学课程考核

命题教师：教研室或系负责人：主管院长：年月日

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《离散数学》课程试题

【A 】卷

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2. Choose the corresponding letter of the best answer that best completes the statements or answers the questions among A, B, C, and D and fill the blanks (3 points each ，15 points in all).

(1) Suppose A = {1, 2, 3}. The following statement ( ) is not true. A ．∅ ⊆ ℘(A ) B ．{∅} ⊆ ℘(A ) C ．{2, 3}∈ A ⨯ A D ．{{2}} ⊆ ℘(A )

(2) Suppose that R and S are transitive relations on a set A . Then ( ) is transitive. A ． S R ⋂ B ．S R ⋃ C ． S R - D ．S R

(3) There are ( ) strongly connected components of the following graph G .

A. 1

B. 2

C. 3

D. 4

(4) There are ( D ) nonisomorphic undirected trees with 5 vertices.

A. 6

B. 5.

C. 4 非同构的无向树

D. 3

(5) Suppose P (x , y ) is a predicate and the universe for the variables x and y is {1,2,3}. Suppose P (1,3), P (2,1), P (2,2), P (2,3), P (3,1), P (3,2) are true, and P (x , y ) is false otherwise. The following statement ( ) is true.

A. ∀y ∃x (P (x , y ) → P (y , x ))

B. ¬∃x ∃y (P (x , y ) ∧ ¬P (y , x ))

C. ∀x ∀y (x ≠ y → (P (x , y ) ∨ P (y , x ))

D. ∀y ∃x (x ≤ y ∧ P (x , y ))

西 南 大 学 课 程 考 核 (试题 【A 】卷)

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3. Write “√” for true, and “⨯” for false in the blanks at end of each statement (3 points each ，15 points in all).

(1) There is a set S such that its power set ℘(S ) has 12 elements. ( ) (2) An irreflexive and transitive relation on a set A is antisymmetric. ( ) (3) The largest value of n for which K n is planar is 6. ( ) (4) Every full binary tree with 61 vertices has 31 leaves. ( ) (5) Logical expressions ))()((x B x A x ∧∀and )()(x xB x xA ∀∧∀are equivalent. ( )

4. For any function f : A → B , define a new function g : ℘(A ) → ℘(B ) as follows: for every S ⊆

A , g (S ) = {f (x )|x ∈ S }. Prove that g is surjective (or onto) if and only f is surjective(or onto). (10 points )

《离散数学》课程试题【A】卷

5.Find the transitive closure t(R) of R on {a, b, c, d} and draw the graph of t(R) where R = {(a, a), (b, a), (b,

c), (c, a), (c, c), (c, d), (d, a), (d, c)}. (10 points)

6. Either give an example or prove that there is none: A graph with 7 vertices that has a Hamilton circuit but no Euler circuit. (10 points)

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西 南 大 学 课 程 考 核 (试题 【A 】卷)

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7. Let G be an undirected tree with 3 vertices of degree 3, 1 vertex of degree 2, the other vertices of degree 1. (15 points )

(1) How many vertices in G are there?

(2) Draw two nonisomorphic undirected trees satisfying the above requirements.