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# 离散数学期末代写 离散数学代写 数学考试助攻 期末考试代考

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## Discrete Mathematics Examination (A)

Id______ Name______ Mark______

### 1. Single-Choice Questions (2×10=20 points).

(1) Among the following four choices, which one is false (≡ denotes logical equivalence)? ( )

A. ﹁(pq)≡p∧﹁q.

B. (pq)∧(pr)≡p→(qr).

C. (pr)∧(qr)≡(pq)→r.

D. (pq)∨(pr)≡p→(qr).

(2) Among the following four choices, which one is false (≡ denotes logical equivalence, and the same domain is used throughout)? ( )

A. ∀x(P(x)∧Q(x))≡∀xP(x)∧∀xQ(x).

B. ∃x(P(x)∨Q(x))≡∃xP(x)∨∃xQ(x).

C. ﹁∀xP(x)≡∃xP(x).

D. ∃x(P(x)∧Q(x))≡∃xP(x)∧∃xQ(x).

(3) Let A and B be two sets, which one is not equivalent to AB? ( )

A. A×BB×B

B. AB=B

C. AB=A

D. AB=∅

(4) Which one has the different cardinality from R (the set of real numbers)? ( )

A. Q (the set of rational numbers).

B. [5, 6].

C. R-[5, 6].

D. RQ.

#### (5) 离散数学期末代写

A = {1, 2, 3}, and R is a binary relation on P(A) (the power set of A), and R={<ab>|aP(A), bP(A), and ab≠∅}. R is .

A. reflexive. B. transitive.

C. symmetric. D. antisymmetric.

(6) Which one is false among the following four statements? ( )

A. An integral ring must be a commutative ring.

B. An integral ring must be a ring with identity.

C. Real number ring R is a field.

D. Z6 with respect to addition modula 6 forms an integral ring.

(7) Let G be a group, and a and b be two finite-order elements of G. Which one is

false among the following choices? ( )

A. |a-1|=|a|.

B. |ab|=|ba|.

C. |b-1ab|=|a|.

D. (ab)n =anbn .

(8) Among the following four pairs of graphs, which pair is not isomorphic? ( )

(9) Among the following four graphs, which one is planar? ( )

(10) Let G be an n-order group, e be its identity, and H be a subgroup of G. Which one is false among the following four propositions? ( )

A. ∀aG, an =e.

B. ∃aG, G=<a>.

C. ∪{aH|aG}=G.

D. ∀aG, there must be a bijective function between H and aH.

### 2. True or False Questions (1×10=10 points).离散数学期末代写

(1) ((pq)∧q)→p is a tautology. ( )

(2) In Boolean Algebra, the Boolean operator ⊕ is also called the XOR operator, defined by 1⊕1=0, 1⊕0=1, 0⊕1=1, and 0⊕0=0. Then x⊕(yz) = (xy)⊕z. ( )

(3) There is a relation R that is both symmetric and antisymmetric. ( )

(4) Let S be a nonempty set. In the poset (P(S), ⊆), there is only one maximal element. ( )

(5) In a lattice, there must be a greatest element. ( )

(6) A 3-order group must be an Abelian group. ( )

(7) Let G be a cyclic group. Any subgroup of G must be a cyclic group. ( )

(8) Let <C, +, ⋅> be an algebraic system, where C is the set of all complex numbers, + and ⋅are common addition and multiplication operators. Then <C, +, ⋅> is a ring. ( )

(9) A nonplanar graph must contain a subgraph homeomorphic to K3,3 or K5. ( )

### 3. Fill in the blank (3×10=30 points).离散数学期末代写

(1) The total number of different Boolean functions of degree 3 is ________ .

(2) Assume that there are 5 Boolean variables: x1, x2, x3, x4, and x5. Write down the minterm ________ that equals 1 if x1=x3=0 and x2=x4=x5=1, and equals 0 otherwise.

(3) Let R1={<1, 5>} be a binary relation on the set A={1, 2, 3, 4, 5}. Then the transitive closure of R1 is ________ .

(4) In the poset ({2, 4, 5, 10, 12, 20, 25}, |), where | denotes divisibility, the maximal elements are ________ .

(5) Let A={1, 2, 3, 4, 5}, and the relation R1 on A be {<1, 2>, <3, 5>}. Then how many equivalence relation R2 satisfying R1R2 can be defined at most on A? ________

(6) Write down all generators of <Z12, ⊕>, ________ , where ⊕ is addition modula 12.

(7) In field Z7, with respect to addition and multiplication modula 7, the solution to the equation 2x=5 is ________ .

(8) How many subgroups are there in <Z6, ⊕> ? where ⊕ is addition modula 6. ________

(9) In the following graph, the total number of the paths of length 3 is ________ .

(10) How many nonisomorphic connected simple graphs are there with 4 vertices? ________

### 4. (10 points) 离散数学期末代写

Use rules of inference to show that if ∀x(P(x) → (Q(x) ∧ S(x))) and ∀x(P(x) ∧ R(x)) are true, then ∀ x(R(x) ∧ S(x)) is true.

### 5. (10 points)

Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d))∈R if and only if a+d=b+c.

(1) Prove that R is an equivalence relation.

(2) Write down the equivalence classes of R.

### 6. (10 points)

Let G be an n-order group, where n is even. Prove that there must be a 2-order element in G.

### 7. (10 points) 离散数学期末代写

Some factory uses 6 colors to paint many pieces of paper. Two sides of each piece of paper need to be painted by two distinct colors, respectively, and such two colors appearing in the same piece of paper are called a pair. Among these pieces of painted paper, every color is combined with at least 3 other colors, respectively, to form pairs to paint different pieces of paper. Using graph theory to prove that among these pieces of paper, we can pick out three pieces of paper, which are painted by the 6 different colors.

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