离散数学模拟题及部分答案(英文)

Discrete Mathematic Test

Editor: Jin Peng

Date: 2008.5.6

Discrete Mathematic Test (Unit 1) (2)

Discrete Mathematic Test (Unit 2) (8)

Discrete Mathematic Test (Unit 3) (13)

Discrete Mathematic Test 1 (17)

Discrete Mathematic Test 2 (22)

Appendix1 Answer to Discrete Mathematic Test(Unit 1) (26)

Appendix2 Answer to Discrete Mathematic Test 2 (31)

Discrete Mathematic Test (Unit 1)

In this part, you will have 15 statements. Make your own judgment, and then put T (True) or F (False) after each statement.

1. Let A, B, and C be sets such that A∪B=A∪C, then B=C. ( )

2. Let A and B be subsets of a set U, and A B, then A△B=A B

and A∩B’=. ( ) 3. Let p a nd q and r be three statements. If ~pÚ~q ≡ ~pÚ~r, then q and r have the same value. ( )

4. Let A, B be sets such that both AÍB and AÎB is possible. ( )

5. Let p and q be two statements, then (p®～q) ®(（～pÚ～q）(p®～q)) is a tautology.

( ) 6.Let A, B be sets, P(A) is the power set of A, then P(A B)=P(A)P(B). ( )

7. Let A, B, and C be sets, then if AÎB，BÍC，then AÍC. ( )

8. Let A, B be sets, if A={Æ}, B=P(P(A)), then {Æ}ÎB and{Æ}ÍB. ( )

9. Let x be real number, then xÎ{x}{{x}} and {x}Í{x}{{x}}. ( )

10. Let A, B, and C be sets, then A(B∪C) = (A B) ∪(A C). ( )

11. If A={x}∪x, then xÎA and xÍA. ( )

12. (x)（P(x)∧Q(x)）and (x)P(x) ∧(x)Q(x) are equivalent. ( )

13. Let A and B be sets, then A×(B C)=(A×B) (A×C). ( )

14. The argument formula (pÚq)® (r s), (sÚt)®w╞ p®w is valid. ( )

15. (x)（P(x) ®Q(x)）and (x)P(x) ® (x)Q(x) are equivalent. ( )

Part II (1 Foundations: Sets Logic, and Algorithms , 85 Scores)

1. (8 points)What sets so each of the Venn diagrams in following Figure represent?

2. (8 points)Let U={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}. Let A={1,5,6,9,10,15} and B={5,6,8,9,12,13}. Determine the following:

Find a. SA b. SA’ c. SB d. SA∩B .

3. (8 points) A class of 45 students has 3 minors for options, respectively A, B and C. A is the set of students taking algebra, B is the set of students who play basketball, C is the set of students taking the computer programming course. Among the 45 students, 12 choose subject A, 8 choose B and another 6 choose C. Additionally, 9 students choose all of the three subjects.

What is the at least number of students do not taking the algebra course and the computer programming course and playing basketball?

4. (8 points)Find a formula A that uses the variables p and q such that A is true only when exactly one of p and q is true.