离散数学期末考试 2013年春季 试卷A - 英才

离散数学期末考试_2013年春季_试卷A- 英才

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电子科技大学英才学院2012-2013学年第2学期期末考试A卷 课程名称：离散数学 考试形式：闭卷考试日期：2013年 %，期中月
日考试时长：120分钟课程成绩构成：平时 1020 %，实验 0 %，期末70 %本试卷试题由____ _部分构成，共_____页。

I.

（）1.

MultipleChoice (15%, 10 questions, 1.5 points each)
Whichof these propositions is not logically equivalent to the other three?a) (p →q) ∧(r →q) b) (p ∨r) →q c) (p ∧r) →q d) ?q →(?p ∧?r)Suppose A ? ?x??y? and B ? ?x???x??, then we don’thave
a)x?? B b)?? P(B). c) ?x? ? A ? B. d)? P(A) ? ? 4.

Supposethe variable x represents students, F(x) means “x is a freshman”,and M(x) means “x is a math major”. Match the statement “??x(M(x)? ?F(x))” with one of the English statements below:

B.Every math major is a
D.Some freshmen are

upto 22?

a) 5

b) 6

c) 7

d) 8

（）9.

（）10.Which of the following is false?

a){x}?{x} b){x}?{x, {x}} c)

{x}?P({x}),where P({x}) is the power set of {x} d){x}?{x, {x}}
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II.True or False (10%, 10 questions, 1 point each)
（）1.The proposition ((p ? q) ? ?p) ? ?q is a tautology.（）2.

（）3.

（）4.

（）5.

（）6.

（）

7.If pigs can fly, then it will be raining tomorrow. Suppose A ??a?b?c?,

then??a?? ? P(A). “My daughter visited Europe last week” implies the

conclusion“Someone visited Europe last week”. For all integers a?b?c?d, if

a??b and c ??d, then (a ? c)?(b ? d). For all real numbers x and y, ?x?

y?? ?x? ? ?y?. h(x)?is defined as a function with domain R and codomainR. （）8..A ? (B ? C) ? (A ? B) ? C.

（）9.The set ????a?????a?? is the power set of some set.

（）10.Let P(m,n) be the statement “m|n,”where the u.d. of m and n is the set of positive integer.

Then?n?mP(m,n)holds.

III.Fill in the Blanks (20%, 10 questions, 2 points each) 1.

2.Suppose A ? ?x ? x ? Z and x2 ? 10???Then P(A) is . ??11If
Ai?{x|x?R???x?}then ?Aiis .iii?1
3.Give a relation on ?a?b?c? that is reflexive and transitive, but notantisymmetric.

.

Supposeg ? A ? B and f ? B ? C where A ? B ? C ? ?1?2?3?4?, g ??(1?4)??(2?1)??(3?1)??(4?2)? and f
??(1?3)?(2?2)?(3?4)?(4?2)?. Then f ? g =
.

Writethe negation of the statement “No tests are easy” in goodEnglish: .

Theexpression of GCD(45?12) as a linear combination of 12 and 45 is
.

permutationsof 7letters A?B?C?D?E?F?G have A immediately to the left of E.

If f (n) ? f (n ? 1) ??f (n ? 2), f (0) ? 2, f (1) ? 5, Then f (2) ?Thenegation of the statement ?x?y (xy = 0) is
4.5. 6. 7. 8. 9.

10.LetR?{(a,b)??2|a?b},R?{(a,b)??2|a?b} 12
ThenR1?R2 is
IV.Answer the Questions (35%,7 questions, 5 points each)：

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1.Write the truth table for the proposition：?(r? ?q) ? (p ? ?r) 2. Suppose f ? R ? R where f(x) ? ?x?2?.

(a)If S ? ?x ? 1 ? x ? 6?, find f(S).

(b)If T ? ?3?4?5?, find f?1(T).

3.Find the matrix that represents the relation of R on ?1?2?3?4? whereaRb means ? a ? b ? ? 1. Use
elementsin the order given to determine rows and columns of the matrix.

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4.

p?q??,and the connective ? that has the given truth table.

5.Encrypt the message “HELP” by translating the letters intonumbers, applying the encryption function
f(p) ? (3p ? 7) mod 26, and then translating the numbers back intoletters.

6.Solve the linear congruence 5x ? 3 (mod 11).

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7.In the questions below suppose g ? A ? B and f ? B ? C where A ? B ?C ? ?1?2?3?4?, g ?

?(1?4)?(2?1)?(3?1)?(4?2)?and f ? ?(1?3)?(2?2)?(3?4)?(4?2)?. Find f ? g. V.(6%) Prove that (q ? (p ? ?q)) ? ?p is a tautology usingpropositional equivalence
andthe laws of logic.

VI.(7%) Let A be the set of all points in the plane with the originremoved.

Thatis,
A= {(x, y) | x, y ∈R} ? {(0, 0)}.

Definea relation on A by the rule: (a, b)R(c, d) ? (a, b) and (c, d) lie onthe same line through the origin. (a) Prove that R is an equivalencerelation. (b)Describe the equivalence classes arising from the equivalencerelation R in part (a).

(c)If A is replaced by the entire plane, is R an equivalence relation?

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VII.(7%) Determine whether this argument is valid: Lynn works part timeor full time.

IfLynn does not play on the team, then she does not work part time. IfLynn plays on the team, she is busy.

Lynndoes not work full time.

Therefore,Lynn is busy.