Download Review sheet

Review sheet
CS 210
Spring 2004
Also look over homework, old exams and the exercises at the end of each chapter. The most important
thing is for you to identify any areas in which you feel weak and get help if you need it.
1. The negation of x P(x) is _______________________
2.. State the inverse, converse and contrapositive of the statement:If m divides n and n divides p, then
m divides p
3. Always, Sometimes or Never: A  (B  C) and B  (A  C) are disjoint.
4. How many elements are in A1  A2  A3 if each set has 90 elements, there are 50
common elements in each pair of sets and 25 elements in all three sets?___________
5. How many subsets of size six does a set with 15 elements have? _______________
6. Suppose A B and C D. Prove or disprove: (A  C)  (B  D).
7. Each of 26 cards has a different letter of the alphabet on it. One card is picked at random. A vowel
is worth 3 points and a consonant is worth 0 points. Find the expected number of points.
8. What is the probability that in 10 rolls of two eight-sided dice an 8 comes up exactly 4 times?
9. Suppose p and q are primes and n = pq. What is the probability that a randomly chosen positive
integer less than n is not divisible by either p or q? How many integers less than n must be chosen
to guarantee that one of them is divisible by p or q?
10. Ten people are wearing badges numbered 1 to 10. Find the probability that if 3 people are
selected at random, the largest number chosen is 5.
11. How many times must a single die be rolled to ensure the same number comes up three times?
12. If a relation is reflexive, its inverse is (always, sometimes, never) reflexive. _________
40
13. Given that 12 + 32 + ... + (2n – 1)2 = n(2n - 1)(2n + 1)/3 , find (2i – 1)2
i=15
14. Construct the truth table for the following statement. Is it a tautology, contradiction, or neither?
q  (p  q)  p
15. A = {1, 2} and B = {a, b}. Show all functions from A to B. For each one, tell if it is one-to-one
and/or onto? How many relations are there from A to B? _______________
16. Prove or disprove: If f  g is onto then g is onto.
17. How many three-digit numbers are there in which the sum of the digits is even?
18. Use mathematical induction to prove that 6 | (n3 + 5n) for n  1.
19. Using the definition of Big O, prove that x2 + 4x + 5 is O(x3).
20. Find the recurrence relation for the amount of money outstanding on a $30,000 mortgage after n
years if the interest rate is 8% and each yearly payment reduces the principal by $3000. State the
initial conditions.
21. Solve the following recurrence: S(n) = 2 S(n-1) + 3 S(n-2) , if S(0) = S(1) = 2.
22. Solve the following recurrence relation: S(n) = 5S(n/2) + 3 with S(1) = 7.
23. Teams A and B are playing a best of 7 match. The probability of A winning is .55, and the probability
of B winning is .45. Assuming the games are independent,
a. Find the probability A will win in four straight games.
b. Find the probability the match will require 7 games to determine a winner.
24. Let A = {a, b, 1} and B = {1, 2, a}.
a. List the elements of (A – B)
b. List the elements of  (A) -  (B)
25. Find the value of k if f(n + 1) = k f(n), f(1) = 5 and f(3) = 20
26. If f(n) = f(n/3) + 2n and f(1) = 1 find f(27). Then, solve the recurrence relation.
27. Let R be a reflexive, transitive relation on a set X. Show that R  R-1 is an equivalence relation
on X.
28. Find the number of bit strings with five 0's and eight 1's that contain the substring 0101.
29. A class contains 15 seniors, 12 juniors and 8 sophomores. How many committees of size 5 contain
a. members of exactly two classes (seniors and juniors, or juniors and sophomores, etc.)
b. exactly three juniors and have a junior as the chair?
30. Prove or disprove that (pq) is logically equivalent to p V q.
31. A relation R is called circular if (a, b)  R and (b, c)  R imply (c, a)  R. Prove that R is
reflexive and circular if and only if it is an equivalence relation.
32. Prove that the function f: R  R defined by f(x) = 2x + 9 is one to one.
33. Give an example of a function f: Z  N that is both one-to-one and onto.
34. Let A = {1, 2, 3, 4, 5, 6, 7} and define relation R on A by (x, y)  R if x - y is a multiple of 3.
a. Show that R is an equivalence relation on A
b. Find the equivalence classes and partition of A induced by R.
35. Suppose R  R-1 = Ø. What conclusion can be drawn about R?
36. Prove or disprove: If Q and R are transitive relations on set S, then Q  R is also transitive.
37. Problems from the text
a. Section 4.1 # 33 a, b (answers are given)
b. Section 4.4 # 8 a – f (answers given)