Exercises
2.2

1. Answer (a)-(c) for the sequences s defined by c, d, d, c, d, c.

a) Find s1 b) find s4
c) Write s as a string

2. Compute the given quantity using the sequence a defined by
an = n2 – 3n + 3.
4 5
a) ∑ ai b) ∑ aj
i=1 j=3

4 6
c) ∑ ai d) ∑ ak
i=4 k=1

3. Answer (a)-(c) for the sequence x defined by
x1 = 2, xn = 3+ xn-1, n ≥ 2

3 10
a) Find ∑ xi b) Find ∑ xi
i=1 i=1

c) Find a formula for the sequence c defined by
n
cn = ∑ xi
i=1

4. Answer (a)-(d) using the sequences y and z defined by
yn = 2n – 1, zn = n (n – 1).

a) Find (3 ∑ yi, i=1) (3 ∑ zi, i=1).
b) Find (5 ∑ yi, i=1) (4∑ zi, i=1).
c) Find 3∑ yi zi.
d) Find (4∑ yi, i=3) (4∏ zi, i=2).



5. Find bi, i = 1, . . . . , 6, where
bn = 2[1 + (n – 1)(n – 2)(n – 3)(n – 4)(n – 5)] + (n – 1)n / 2..

6. List the members of these sets.
a) {x x is real number such that 12x2'> = 1}
b) {x x is a positive integer less than 12}
c) {x x is the square of an integer and x ˂ 100}
d) {x x is an integer such that 12x2'> = 2}

7. For each of the following sets, determine whether 2 is an element of the set.
a) {x Є R is an integer than 1}
b) {x Є R is the square of an integer}
c) {2{2}}
d) {{2}, {2{2}}
e) {{2},{2, {2}}}
f) {{{2}}}

8. Determine whether each of these statements is true or false.
a) 0 Є
b) = {0}
c) {0}
d) {0}
e) {0} Є {0}
f) {0} {0}
g) { } Є { }

9. Determine whether each of these statements is true or false.
a) X Є {x}
b) {x} {x}
c) {x} Є {x}
d) {x} Є {{x}}
e) {x}
f) Є {x}

10. What is the cardinality of each of these sets?
a) {a}
b) {{a}}
c) {a, {a}}
d) {a, {a}, {a, {a}}}

11. Find the power set of each of these sets.
a) {a}
b) {a, b}
c) { , { }}

12. Let A = {a, b, c, d} and B = {y, z}. find
a) A x B
b) B x A

13. What is the Cartesian product A x B x C, where A is the set of all airlines and B and C are both the set of all cities in the United States.

14. How many different element does A x B have if A has m element and B has n elements?

15. Show that A x B B x A, when A and B are nonempty, unless A = B.

16. Translate each of these qualifications into English and determines its truth value.
a) x Є R ( 12X2'> -1)
b) x Є 2 ( 12X2'> > 0)
c) Ǝ x Є Z ( 12X2'> = 2)
d) Ǝ x Є R ( 12X2'> = x)

2.1 Sets

Exercises

1. Let A be the set of students who lived within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets.
a) A ∩ B b) A U B
c) A – B d) B – A
3. Let A = {1, 2, 3, 4, 5} and B = {a, b, c, d, e, f, g, h}. Find
a) A U B b) A ∩ B
c) A – B d) B – A
4. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find
a) A U B b) A U B
c) A – B d) B – A
9. Let A and B be sets. Show that A ∩ (A U B) = A.
19. Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9}. Find
a) A ∩ B ∩ C b) A U B U C
c) (A U B) ∩ C d) (A ∩ B) U C