Chapter 1

Question 1

Element

Answer 1

A element is an object

Question 2

Null set

Answer 2

A null set is a set with no elements

Question 3

Well-defined set

Answer 3

A set is well-define if for each element we can decide if it is in or if it is not in the set

Question 4

Universal set

Answer 4

All elements under consideration

Question 5

Complement

Answer 5

Let A be a set; U be the universal set. Ac is everything in the U that’s not in A.

Question 6

Subsets

Answer 6

Every element of A is also an element of B

Question 7

Equality

Answer 7

A=B off A is a subset of B and B is a subset of A

Question 8

Relative complement

Answer 8

A\B

Question 9

Disjoint

Answer 9

A and B are disjoint if the intersection of A and B is the empty set.

Question 10

Demorgan’s laws

Answer 10

If A and B are sets, the

a) (AUB)c=Ac and Bc
b) (A and B)c= AcUBc

Question 11

Functions

Answer 11

Let A and B be sets. A functions, f, from A to B is a rule which associates each element x in A with a unique element f(x) in B.

Question 12

Domain

Answer 12

Let f:A->B. A is the domain of f

Question 13

Range

Answer 13

Let f:A->B. The set of elements in B that have some point of A mapped to them by f is called the range of f.

Question 14

Equal functions

Answer 14

Two functions f and g are equal if

1) domain of f=domain of g
2) f(x)=g(x) for all c in their common domain.

Question 15

One to one

Answer 15

F is a one to one function if different elements of A are mapping to different elements of B.
If x1,x2 in A with x1 does not equal x2, then f(x1) does not equal f(x2)
If x1, x2 in A with f(x1)=f(x2), then x1=x2

Question 16

Onto

Answer 16

F is onto if for all y in B, there exist an x in A such that f(x)=y.
“f uses all of B”

Question 17

Inverse

Answer 17

Let f:X->Y be a one to one function. Then the function f-1:R(f)->X defined by f-1(y)=x provided y=f(x) is call the inverse of the function f.

Question 18

Field

Answer 18

A field,F, is a non empty set with two operations, + and -, which satisfys the following
  Binary
  Associative
  Commutative
  Distribution works
  Identities
  Additive inverse
  Multiplication inverse

Question 19

The order axiom

Answer 19

There is a nonempty subset P(positives) of F for which

1) a,b in P implies a+b is in the positives
2) a,b in P implies a*b is in the positives
3) for any x in F, exactly ONE of the filling holds: x is positive, x is negative , or x is zero.

Question 20

Order field

Answer 20

A field satisfying the order axiom is an ordered field.

Question 21

Interval

Answer 21

An interval of real numbers is a set, A, containing at least 2 numbers such that if r,s are in A with r

Question 22

Upper bound

Answer 22

If there exist a b in R st x

Question 23

Bounded above

Answer 23

A set is bounded above if it has an upper bound.

Question 24

Lower bound

Answer 24

If there exist a c in R such that c>=x for all x in A. Then A had a lower bound.

Question 25

Bounded below

Answer 25

A set is bounded below if it has a lower bound.

Question 26

Bounded

Answer 26

If a set is bounded above and below.

Question 27

Unbounded

Answer 27

If it is not bounded

Question 28

Least upper bound

Answer 28

Let A be a set of real numbers that is bounded above. The number b is called the least upper bound of A if b is an upper bound of A. If k is also an upper bound of A, then b

Question 29

Supremun

Answer 29

Least upper bound

Question 30

Greatest lower bound

Answer 30

c is a lower bound of A. If l is a lower bound of A, then c>=l.

Question 31

Imfimum

Answer 31

Greatest lower bound

Question 32

Complete

Answer 32

Let S be an ordered field. Then S is complete if for any nonempty subset A of S that is bounded above, lub(A) is in S.

Question 33

Completeness axiom

Answer 33

R is complete

Question 34

The Archimedean principle

Answer 34

If a,b are in R with a>0, then there exist an n in the natural numbers such that na >b

Question 35

Cardinal its

Answer 35

Two sets A and B have the same cardinality if there is a one to one, onto function from the set A to the set B.

Question 36

Countable

Answer 36

A set is countable if it has the same cardinality as some subset of the natural numbers

Question 37

Set

Answer 37

A set is a collection of objects

Question 38

Cartesian product

Answer 38

Let A1,A2,…An be a finite collection of sets. The Cartesian product of the collection is A1xA2x…xAn={(a1,a2,….,an)|aj is in Aj for j=1,2,3….,n}

Ex: A={1,2,4} and B={3,5,6}
The AxB={(1,3),(1,5),(1,6),(2,3),(2,5), ect

Question 39

Triangle inequality

Answer 39

|a+b|