Chapter 0

Question 1

the union of a collection of sets {Bα : α ∈ Λ}

Answer 1

U {Bα : α ∈ Λ} = UB. There exists an α that is a member of Bα

Question 2

the intersection of a collection of sets {Bα : α ∈ Λ}

Answer 2

The intersection of {Bα : α ∈ Λ}. There is a member of Bα for all α

Question 3

DeMorgan’s Laws 1. Let U be a set, let {Bα : α ∈ Λ} be a collection of sets. The complement of the union is the

Answer 3

intersection of the complement

Question 4

DeMorgan’s Laws 2. Let U be a set, let {Bα : α ∈ Λ} be a collection of sets. The complement of the intersection is the

Answer 4

union of the complement

Question 5

Let A and B be sets, and let f : A → B. Let S ⊆ B. The inverse image of a set S ⊆ B under the function f : A → B is f-1[S]=

Answer 5

{a ∈ A: there exists b ∈ S st f(a)=b}

Question 6

Let A and B be sets, and let f : A → B. Let T ⊆ A. The image of a set T ⊆ A under the function f : A → B is f[T]=

Answer 6

{f(t): t ∈ T}

Question 7

totally ordered set

Answer 7

Let A be a set. We say that A is a ______ if there is a relation ≤ on A that satisfies the following properties:

1. For all a ∈ A, a ≤ a.
2. If a, b ∈ A, then either a ≤ b or b ≤ a.
3. If a, b ∈ A with a ≤ b and b ≤ a then a = b.
4. If a, b, c ∈ A with a ≤ b and b ≤ c, then a ≤ c.

Question 8

Let A be a non-empty set. Any function s : N → A is called a _____ in A

Answer 8

sequence (sn)

Question 9

Let (sn) be a sequence in a totally ordered set A.

1. (sn) is a sequence of distinct terms if si

Answer 9

does not = sj whenever i does not = j.

Question 10

Let (ni) be a strictly increasing sequence of natural numbers. Then,

Answer 10

Then i ≤ ni for all i ∈ N.

Question 11

Let A be a set. Let (sn) be a sequence in A. If (ni) is a strictly increasing sequence in N, then the sequence
sn1, sn2, sn3, . . .is a _________. We denote this by (sni).

Answer 11

subsequence of (sn)

Question 12

There are three pieces of information in the notation (sni)

Answer 12

1. sni is an element of A and gives the value of the term
2. ni is a natural number and indicates the position of the term in the original sequence
3. i is also a natural number and indicates the position of the term in the subsequence.

Question 13

Complement of B in U

Answer 13

Let B⊆U. {x∈U: x/∈ B}

U\B

Question 14

Theorem 0.2.7 about properties of an inverse. Let A and B be sets. Let f: A → B. TFAE

Answer 14

1. f is injective and onto (bijective)
2. There exists a function f-1: B→A st it has the properties of an inverse
   (a) The composition of f inverse and f(x) is x
   (b) the composition of f and f inverse(x) is x

Question 15

Theorem 0.2.12 about families of subsets and inverses. Let A and B be sets and f: A → B be a function. Let {Sα : α ∈ Λ} be a family of subsets of B. Then 1. The inverse image of the union of Sα is equal to

Answer 15

the union of the pre-image of the Sα.

Question 16

Theorem 0.2.12 about families of subsets and inverses. Let A and B be sets and f: A → B be a function. Let {Sα : α ∈ Λ} be a family of subsets of B. Then 2. the inverse image of the intersection is equal to

Answer 16

the intersection of the pre-image of Sα

Question 17

Theorem 0.2.12 about families of subsets and inverses. Let A and B be sets and f: A → B be a function. Let {Sα : α ∈ Λ} be a family of subsets of B. Then 3. The complement of the inverse image is equal to

Answer 17

the pre-image of the complement of Sα

Question 18

Theorem 0.2.16 about families of subsets and their unions and intersections. Let A and B be sets and f: A → B. Let {Tα:α ∈ Λ} be a collection of subsets of A. Then, the image of the union (f[UTα]) is equal to

Answer 18

the union of the image of Tα (Uf[Tα])

Question 19

Theorem 0.2.16 about families of subsets and their unions and intersections. Let A and B be sets and f: A → B. Let {Tα:α ∈ Λ} be a collection of subsets of A. Then, the image of the intersection (f[UTα]) is contained in

Answer 19

the intersection of the image of Tα (Uf[Tα])

Question 20

Totally Ordered Set (Trichotomy Law)

Answer 20

For all x and y in A:

1. x>y or
2. x is less than y
3. x equals y

Question 21

The ________ of the ________ is contained in the intersection of the images.

Answer 21

image of the intersection

Question 22

2. (sn) is a constant sequence if

Answer 22

there exists a ∈ A such that si = a for all i ∈ N.

Question 23

3. (sn) is increasing(resp. decreasing) if

Answer 23

whenever n ≤ m, sn ≤ sm (resp. sn ≥ sm)

Question 24

4. (sn) is strictly increasing (resp. strictly decreasing)

Answer 24

if whenever n sm)

Question 25

5. (sn) is monotonic if

Answer 25

it is either increasing or decreasing.

Question 26

6. (sn) is bounded from below (resp. bounded from above) if

Answer 26

there exists b ∈ R such that for all i ∈ N, b ≤ si (resp. b ≥ si). In this case, b is called a lower bound (resp. upper
bound) for (sn).

Question 27

7. If (sn) is bounded from above and from below, then

Answer 27

it is bounded