Math 141 Test 1 Carter Name________________________
Show all work. 2/7/07
1. A = { 1, 3, 6, 8 } B = { 3, 4, 8 } C = { 1, 4 } U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
a) AC = b) A-B = c)
C
= d) B x C = (12)
2. True or False (12)
____ a) A B A ____ b)
A B A B
____ c) (AB)C =(AC)B
____ d) A x B = B x A ____ e) 3 7 (mod 4) ____ f) 8 -3 (mod 5)
3. Use a Venn diagram to illustrate
A B
. (4)
4. Does
A B A C
imply that B = C ? Justify your response. (3)
5. Suppose
B
, under what conditions, if any, is A B = A? (3)
6. Using theorems from chapter 2, simplify
A B A
so that the expression contains at most one set operation.
You may use diagrams to check, but you will not get full credit for this method. (6)
7. How many subsets does {red, blue, yellow, green} have? (4)
8. Let S be the set of CBU students. Consider a relation R defined on S. Determine which of the reflexive,
symmetric, and transitive properties are satisfied when we define xRy to mean that the height of x differs from the
height of y by no more than one inch. Give reasons for your choices. Is R an equivalence relation? If it is,
describe the equivalence classes.
(6)
9. Write the equivalence relation on {1, 2, 3, 4} that is induced by the partition with {1, 3, 4} and {2} as its partitioning
subsets. (6)
10. Find the quotient and remainder in the division of n = -57 by m = 11. (4)
11. Perform the indicated calculations in Zm. Write you answer in the form [r] with 0 r < m.
(9)
a) [23]+[15] in Z6b) [16][34] in Z5c) [18]8 in Z7
12. In Z8, which of the following congruence classes are equal: [2], [7], [10], [16], [39], [-6]? (6)
13. Give an example to show that in Z8 it is possible for [x] [0] and [y] [0], but [x][y] = [0]. (4)
14. Recall that the check digit on an ISBN is determined by multiplying the first nine digits by 10, 9, 8, 7, 6, 5, 4, 3, and
2 respectively, and adding these nine products to obtain a number y. The check digit d is then chosen so that
y + d 0 (mod 11). Determine the correct check digit for the ISBN that has 0-618-56806 as its first nine digits.
(6)
15. a) Define subsets, X and Y, of the real numbers so that
: ( ) 2
x
g X Y where g x
is one-to-one & onto fn.
b) X is the set of students currently enrolled at CBU University, and for xX, h(x) is the year that x first took a
computer science course at CBU.
Is h(x) a function? _________ one-to-one? ______
(8)
16. Prove one of the following. (8)
a) (A-B) (B-A) = (AB) - (AB) (Only partial credit for diagram.)
b) Use the definition of congruence to show that if a b (mod m) and b c (mod m) then a c (mod m).
