406
Relations, Functions, and Matrices
26. Let S = 5a, b, c, d6 and T = 5x, y, z6.
a. Give an example of a function from S to T that is neither onto nor one-to-one.
b. Give an example of a function from S to T that is onto but not one-to-one.
c. Can you find a function from S to T that is one-to-one?
27. Compute the following values.
a. :3.4;
b. <−0.2=
c. :0.5;
28. Compute the following values.
a. <−5−1. 2=
b. <−5−<1.2= =
c. :2 * 3.7;
d. <1 + 1∙2 + 1∙3 + 1∙4=
29. What can be said about x if :x; = 30. Prove that 31. Prove that :x; = −<−x=.
32. The ceiling function f (x) = a. f is one-to-one
b. f is onto
33. Prove or disprove:
a. < :x;= = x
b. :2x; = 2:x;
34. Prove or disprove:
a. :x; + : y; = :x + y;
b. :2x; = :x; + :x + 1∙2;
35. Prove that if 2
k
< n < 2
k+1
then k = :log n; and k + 1 = 36. Prove that if 2
k
≤ n < 2
k+1
then :log n; + 1 = 37. Compute the value of the following expressions.
a. 31 mod 11
b. 16 mod 8
c. 22 mod 6
d. −7 mod 3
38. a. List five values x such that x mod 7 = 0.
b. List five values x such that x mod 5 = 2.
39. Prove or disprove: For any integers x and y, x mod 10 + y mod 10 = (x + y) mod 10.
40. Prove that x ≡ y (mod n) if and only if x mod n = y mod n. (Recall the definition of congruence modulo
n from Section 5.1.)
41. Let S be a set and let A be a subset of S. The characteristic function of A is a function c A : S S 50, 16 with
c A (x) = 1 exactly when x [ A.
a. Let S = 51, 2, 3, 4, 56 and A = 51, 3, 56. Give the ordered pairs that belong to c A .
b. Prove that for any set S and any subsets A and B of S, c A d B (x) = c A (x) # c B (x).