CS 210 Review Sheet
Answers.
1. What is the negation of x P(x)?
Fall 2011
x (P(x))
2. State the inverse, converse and contrapositive of the statement: “If m divides n and n divides
p, then m divides p”.
Inverse: If m does not divide n or n does not divide p, then m does not divide p
Converse: If m divides p then m divides n and n divides p
Contrapositive: If m does not divide p then m does not divide n or n does not divide p
3. Always, Sometimes or Never: A  (B  C) and B  (A  C) are disjoint.
4. What is |A  B  C| if |A| = |B| = |C| = 90, |AB| = |AC| = |BC| = 50 and |ABC| = 25?145
5. How many subsets of size six does a set with 15 elements have? C(15, 6)
6. How many strings of length six can be formed from a 15 letter alphabet?
a. with repetition allowed? 156
b. with all letters distinct? P(15, 6)
7. Suppose A B and C D. Prove or disprove: (A  C)  (B  D).
Let x  A  C. then x  A and x  C by definition of intersection.
Since x  A and A  B, then x  B by definition of subset.
Since x  C and C  D, then x  D by definition of subset.
Therefore, x  B D
8. A sailing club has 10 white flags, 7 red flags, and 3 green flags. If all flags are displayed on
a flag pole, how many different arrangements are possible? 20!/(10!•7!•3!)
9. How many five card poker hands have exactly 2 pairs? C(13, 2)C(4, 2)C(4, 2)44
10. Let S = {1,2,3,4,5,6,7,8,9,10}. In how many 5-element subsets of S is 3 the smallest element?
C(7, 4) There are seven elements greater than three of which four must be chosen to
obtain a subset of size five that includes 3.
11. A five-card hand is selected from a deck of 52 cards. How many hands contain no aces or kings?
C(44, 5); exactly 2 queens? C(4, 2)•C(48, 3); exactly 2 queens and exactly 3 spades?
C(3, 2)•C(12, 3) + C(3, 1)•C(12, 2)•36 (without and with the Q of spades)
12. A string of length 8 is formed from the alphabet A = {a, b, c, d, e, f}. How many strings contain
exactly one a or exactly two b’s?
(exactly 1 a +
8•57
+
exactly 2 b’s –
C(8,2)•56
-
1a and 2b’s)
8•C(7, 2)•45
13. A pizza take-out advertises it offers a choice of over 200 different pizzas. You can choose
one or more different toppings. How many toppings must be available to make this
statement true? 8 since 28 = 256
14. How many distinct permutations are there of the letters in COOPERATE? 9!/ (2!•2!)
15. f: R  R, what is the range of the function f(x) = 1/(x2 + 4)? all y such that 0 < y  ¼
n
16. Write the following using  notation n + (n-1)/2! + (n-2)/3! + (n-3)/4! + … + 1/n!  (n-i+1)/i!
i=1
40
17. Given that 12 + 32 + ... + (2n – 1)2 = n(2n - 1)(2n + 1)/3 , find (2i – 1)2 40·79·27 - 14·9·29
i=15
18. Construct the truth table for the following statement. Is it a tautology, contradiction, or
neither? q  (p  q)  p
This is a tautology
p q pq q q  pq p
q  pq  p
T T
T
F
F
F
T
T F
F
T
F
F
T
F T
T
F
F
T
T
F F
T
T
T
T
T
19. A = {1, 2} and B = {a, b}. Show all functions from A to B. For each one, tell if it is oneto-one and/or onto. How many relations are there from A to B?
f1(1) = a
f2(1) = a
f3(1) = b
f4(1) = b f2 and f3 are one-to-one and onto
f1(2) = a
f2(2) = b
f3(2) = a
f4(2) = b There are 24 relations.
20. Prove or disprove: If f  g is onto then g is onto. Not true
Let A = {1, 2}, B = {3, 4, 5} and C = {a} where g(1) = g(2) = 3 and f(3) = f(4) = f(5) = a
21. Use mathematical induction to prove that 6 | (n3 + 5n) for n  1.
Basis: n = 1, n3 + 5n = 1 + 5´1 = 1 + 5 = 6 and 6 | 6 so P(1) holds.
Hypothesis: 6 | (n3 + 5n) for n  1
Induction Step: (n+1)3 + 5(n+1) = n3 +3n2 +3n + 1 + 5n + 5 =
n3 + 5n + 3(n2 + n + 2)
By hypothesis 6 | n3 + 5n
3 | 3(n2 + n + 2) and (n2 + n + 2) is even, so 6 | 3(n2 + n + 2)
Therefore, 6 | n3 + 5n + 3(n2 + n + 2)
22. Using the definition of Big O, prove that (x2 + 4x + 5)  O(x3).
x2 + 4x + 5  x3 + 4x3 + 5x3 = 10x3
Thus, let K = 10 and N = 1. Then, by the definition of big-O, x2 + 4x + 5 = O(x3)
35
23. Evaluate the following summation  (4i – 2)
(436 – 410)/3 – 2•26
i=10
24. Solve the following recurrence f(n) = 8f(n/2) + 3 and f(1) = 3. Assume n = 2k. 3n3 + 3(n3-1)/7
25. How many bit strings of length 8 begin with 101 or have a 1 as the fourth bit? 25 + 27 - 24 = 144
26. A binary relation R on the set of integers is defined by (a, b)  R iff ab  0. Prove or
disprove: R is an equivalence relation. What is R-1?
For any integer a, a•a  0, so (a, a)  R and R is reflexive.
If (a, b)  R then a•b  0. Thus, b•a  0 and (b, a)  R and R is symmetric.
R is not transitive since (2, 0)  R and (0, -5)  R but (2, -5)  R since 2(-5) = -10
(a, b)  R-1 iff ba  0 so R-1 = R
27. Find the value of k if f(n + 1) = k f(n), f(1) = 5 and f(3) = 20
k = 2 or k = -2
28. If f(n) = f(n/3) + 2n and f(1) = 1 find f(27). f(27) = 79
29. Let R be a reflexive, transitive relation on a set X. Show that R  R-1 is an equivalence relation.
R  R-1 is reflexive since a  X, (a, a)  R. By the definition of inverse, (a, a)  R-1
and thus (a, a)  R  R-1. If (a, b)  R  R-1 then (a, b)  R and (a, b)  R-1. Therefore,
by the definition of inverse, (b, a) is in both R and R-1 and thus R  R-1 is symmetric.
Finally, suppose both (a, b) and (b, c) are in R  R-1. Then, both (a, b) and (b, c) are
in R and also in R-1. Since R is transitive, (a, b) and (b, c) in R imply (a, c)  R. By
the definition of inverse, (b, a) and (c, b) are in R. Since R is transitive, (c, a)  R and
thus (a, c)  R-1. This means (a, c)  R  R-1 and thus R  R-1 is transitive.
30. Find the number of bit strings with five 0's and eight 1's that contain the substring 0101.
10•C(9, 3) (Arrange 10 things: the 0101 pattern + three 0's and six 1's)
31. A test has two parts. Part 1 has 6 questions and part 2 has 4. A student must answer a
total of 6 questions with at least two from each part. In how many ways can the questions
that are answered be selected? C(6, 2)•C(4, 4) + C(6, 3)•C(4, 3) + C(6, 4)•C(4, 2)
32. Find an integer a > 7 such that a  3a (mod 7). One possible answer is a = 14.
33. Prove or disprove: If a  b (mod m), and if c  d (mod m) then ac  (b + d) (mod m)
False. Here's one counterexample: a = 5, b = 5, c = 2, d = 2, and m = 5
34. Prove that (pq) is logically equivalent to p V q.
q
p V q
p
q
pq pq) p
T
T
T
F
F
F
F
T
F
F
T
F
T
T
F
T
F
T
T
F
T
F
F
F
T
T
T
T
35. Prove or disprove: if R1 and R2 are equivalence relations, then so is R1  R2
Not necessarily since the union may not be transitive. Let S = {1, 2, 3} and
R1 = {(1,1), (2, 2), (3, 3), (1,2), (2, 1)} and let R2 = {(1,1), (2, 2), (3, 3), (2,3), (3, 2)}.
To be transitive (1,3) would have to be in the union.
36. For integers a, b define a relation R on A by (a, b)  R iff 2a+ 3b = 5n for some integer n.
Prove that R is a reflexive, symmetric relation on Z. (It’s also transitive but that proof is a bit
messy.)
Given a  A, 2a + 3a = 5a, so (a, a)  R and R is reflexive
If (a, b)  R then 2a + 3b = 5k so b = (5k – 2a)/3 which is an integer. Now, consider
2b + 3a = 2((5k – 2a)/3) + 3a where both terms are integers so the sum is an integer.
Simplifying and putting everything over a common denominator we get (10k – 4a +
9a)/3 = 5(2k + a)/3 so (b, a)  R and R is symmetric.
37. What properties does the following relation possess?
R = {((a, b), (c, d)) | a + b  c + d} For example, ( (2, 3), (4, 1) )  R That is, the elements of R
are pairs of ordered pairs
Reflexive ( (a,b), (a,b) )  R for all pairs (a, b)
Antisymmetric If a + b  c + d and c + d  a + b then a + b = c + d
Transitive If ( (a,b), (c,d) ) and ( (c,d), (e,f) )are in R then a + b  c + d and c + d  e + f,
then a + b  e + f. Therefore ( (a,b), (e,f) )  R so R is transitive.
38. If a relation is reflexive, its inverse is (always, sometimes, never) reflexive.
39. If a relation is reflexive, its complement is (always, sometimes, never) reflexive.
40. Suppose |A| = n and that R is an antisymmetric relation on A. What is maximum value of |R|?
(n2 + n)/2—a largest antisymmetric relation will be obtained by choosing all ordered pairs
above and on the diagonal.
41. If R = {(1, 3), (2, 1), (2, 3), (3, 2), (4, 2)} is a relation on S = {1, 2, 3, 4}, find the transitive
closure of R.
{(1, 3), (2, 1), (2, 3), (3, 2), (4, 2), (1, 2), (2, 2), (3, 1), (3, 3), (4, 1), (4, 3), (1, 1)}
42. The set { {1, 3, 5}, {2}, {4, 6} } is a partition of S = {1, 2, 3, 4, 5, 6}. List all ordered pairs in the
corresponding equivalence relation.
{(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5), (2, 2), (4, 4), (4, 6), (6, 4), (6, 6)}
43. For each relation below, place its number in the blank to the left of each property it satisfies.
Relation 1: R = {(a, b) N  N | a + b is even}
Relation 2: R = {(a, b) | a, b  Z and |a - b|  1}
Relation 3: R = 
_____1, 2_____ Reflexive ___1, 2, 3_____ Symmetric __1, 3_______ Transitive
______3______ Irreflexive _____3_____ Antisymmetric ____3______ Asymmetric