FOM Test 1

Using bar notation, describe the set of rational numbers between 0 and 1. Then describe the set of positive rationals whose denominator is a power of 2, such as 7/2, 3/4,5or 1/16 . (The powers of 2 are 1,2,4,8,....)

The first set is {m/n | m,n ∈ N and m < n}. The second set may be described as{m/2(power of)k | m, k ∈Z and m≥1, k≥0}.

Define a universal set U = {a, b, c, d, e, f, g, h}. Using these elements, con- struct two sets A and B satisfying |A| = 5, |B| = 4 and |A∩B| = 2. Using the sets you chose, compute |-A ∩ -B|.

One possibility is A = {a, b, c, d, e} and B = {d, e, f, g}. Regardless of your choice, you should find that |-A ∩ -B| = 1

Let A = {x | 1 < x < 3}, B = {x | 5 ≤ x ≤ 7} and C = {x | 2 < x < 6}, where x represents a real number. Determine the sets A ∪ C, (A ∪ B) ∩ C and B ∩ -C, writing your answers in bar notation.

A∪C={x|1<x<6},(A∪B)∩C={x|2<x<3or5≤x<6} and finally B ∩ -C = {x | 6 ≤ x ≤ 7}

Why is it not possible for two sets to satisfy both A ∩ B = {f,o,u,r} and A ∪ B = {f, o, r, t, y, s, i, x}?

The first statement requires that u∈A and u∈B. So must have u∈A∪B.

Let A and B be the sets of students in a certain class who are sophomores and who are from New York, respectively. Write an expression that represents the set of students who are sophomores or who come from outside New York.

The set is A∪B. (line over B)

LetB={I2m+5n| m,n∈N}. Is 10∈B? Is 13∈B? Explain.

We have 10 ̸∈ B but 13 ∈ B

List the four possible ways that x could be (or not be) an element of two given sets A and B. In each case identify the corresponding region in the labelled Venn diagram within this section.

We have x∈A and x∈B(regionII),x∈A and x̸∈B(regionI),x̸∈A and x ∈ B (region III), x ̸∈ A and x ̸∈ B (region IV )

Briefly justify why the following statements are true or false. a) If A is the set of letters in the word 'flabbergasted,' then |A| = 13. b) For the set A in the previous part, we have a ∈ A or z ∈ A. c)If B={n|n∈Z, 10≤n≤20} then |B|=10. d)For the set B in the previous part we have 11∈B and √200∈B. e) If L is the set of letters in your full legal name, then a ∈ L. f) Let C = {x | x ∈ R, x2 ≤ 10}. If π ̸∈ C then −3 ̸∈ C.

a) False b) True c) False d) False e) Depends f) True

Given the universal set U = {a,b,c,...,z}, we define A = {b,r,i,d,g,e}, B = {f,o,r,t,y,s,i,x} and C = {s,u,b,z,e,r,o}. Decide whether the following statements are true or false. a) |A ∪ C| = 10 b) B ∩ -B = ∅ c) |B ∪ -C| = 23 d) (A ∪ B) ∩ C = {s, o, b, e, r} e) |(A ∩ B) ∪ (B ∩ C)| = 5 f) -A ∩ -B ∩ -C = {a, c, h, j, k, l, m, n, p, q, v, w, y} g) |A ∪ B ∪ C| = 15

a) True b) True c) False d) True e) False f) False g) False

Describe the following sets using bar notation. a) A is the set of all integers divisible by 7 b) B = {3,5,9,17,33,65,···} c) C is the set of all real numbers between √2 and π d) D = { 1/2 , 1/3 , 1/4 , 1/5 , . . .}

a)A={7k|k∈Z} b)B={2n+1|n∈N} c)C={x|x∈R, √2<x<π} d) D = {1/(m + 1) | m ∈ N}