Chapter 1 (part a)

Question 1

Field

Answer 1

Any set where addition and multiplication are well-defined operations that are communative, associative, distributive, have an additive identity, multiplicitive inverse, and multiplicitive idenitity.

Question 2

Communitive Property

Answer 2

Changing the order of the operands does not change the result.

Question 3

Associative Property

Answer 3

Rearranging the parenthesis does not change result

Question 4

Distributive Property

Answer 4

A sum times a value may be expanded in terms the sum of each component times the value

Question 5

Additive Identity

Answer 5

For some value x, the additive identity plus x yields x

Question 6

Multiplicative Inverse

Answer 6

A number x, when multiplied by a its multiplicative inverse, yields 1

Question 7

Multiplcative Identity

Answer 7

The number 1, from any value times its multiplicative inverse is 1. Also ay value times the multiplicative identity is the value

Question 8

Set

Answer 8

A collection of elements

Question 9

Disjoint

Answer 9

the intersection of two sets A and B is the empty set

Question 10

Set B contains set A

or A is a subset of B

Answer 10

A c B

Question 11

U_{n=1 to inf} (A_n)

Answer 11

The infinite intersections of the sets A_i

=A₁UA₂UA₃….

Question 12

De Morgan’s Laws

Answer 12

(AΠB)^c = A^cUB^c

and

(AUB)^c = A^cΠB^c

Question 13

Function

Answer 13

Rule or mapping that takes each a in A and associates it with a single element of B

Question 14

Triangle Inequality

Answer 14

|a+b| < |a| + |b|

Question 15

Theorem: Two real numbers a and b are equal if and only if…

Answer 15

For every real number ϵ > 0, it follows that |a - b| < ε

Question 16

Axiom of Completeness

Answer 16

Every non-empty set of real numbers, that is bounded above, has at least one upper bound

Question 17

Bounded Above

Answer 17

A set A c R is bounded _____ if there exists a b in B such that

a < b for all a in A.

And b is an ______ bound for A.

Question 18

Bounded Below

Answer 18

A set A c R is bounded ____ if there exists b in R such that

b < a for all a in A.

And b is an ____ bound for A.

Question 19

Least Upper Bound

for some b in R, b is the least upper bound if for a set A c R if:

Answer 19

i. ) b is an _____ bound of A
ii. ) if c is any ____ bound of A, b < c

Question 20

Greatest Lower Bound

for some b in R, b is the greatest lower bound if for a set A c R if:

Answer 20

i. ) b is an _____ bound of A
ii. ) if c is any ____ bound of A, b > c

Question 21

One to One

Answer 21

a function f:A to B is one-to-one if a₁≠a₂ and f(a₁)≠f(a₂)

Question 22

Onto

Answer 22

f is onto if given b in B, there exists a in A such that f(a)=b

Question 23

Pre-image

Answer 23

Given a function f:D to R and a subset B c R, let _____ be the set of all points from the domain D that get mapped into B

____={x in D: f(x) in B}

Question 24

Theorem- Assume s in R is an upper bound for a set A c R. Then s=sup(A) if and only if, for every choice of ε > 0….

Answer 24

there exists some a in A satisfying s - ε

Question 25

Theorem- Assume i in **R** is a lower bound for a set A _c_ **R**. Then i = inf(A) if and only if for every choice of ε \> 0....

Answer 25

there exists some a in A such that i + ε \> a

Question 26

Any set where addition and multiplication are well-defined operations that are communative, associative, distributive, have an additive identity, multiplicitive inverse, and multiplicitive idenitity.

Answer 26

Field

Question 27

Changing the order of the operands does not change the result.

Answer 27

Communitive Property

Question 28

Rearranging the parenthesis does not change result

Answer 28

Associative Property

Question 29

A sum times a value may be expanded in terms the sum of each component times the value

Answer 29

Distributive Property

Question 30

For some value x, the additive identity plus x yields x

Answer 30

Additive Identity

Question 31

A number x, when multiplied by a its multiplicative inverse, yields 1

Answer 31

Multiplicative Inverse

Question 32

The number 1, from any value times its multiplicative inverse is 1. Also ay value times the multiplicative identity is the value

Answer 32

Multiplcative Identity

Question 33

A collection of elements

Answer 33

Set

Question 34

the intersection of two sets A and B is the empty set

Answer 34

Disjoint

Question 35

A _c_ B

Answer 35

Set B contains set A or A is a subset of B

Question 36

The infinite intersections of the sets A_i =A₁UA₂UA₃....

Answer 36

U_{n=1 to inf} (A_n)

Question 37

(AΠB)^c = A^cUB^c and (AUB)^c = A^cΠB^c

Answer 37

De Morgan's Laws

Question 38

Rule or mapping that takes each a in A and associates it with a single element of B

Answer 38

Function

Question 39

|a+b| _\<_ |a| + |b|

Answer 39

Triangle Inequality

Question 40

For every real number ϵ \> 0, it follows that |a - b| \< ε

Answer 40

Theorem: Two real numbers a and b are equal if and only if...

Question 41

Every non-empty set of real numbers, that is bounded above, has at least one upper bound

Answer 41

Axiom of Completeness

Question 42

A set A _c_ **R** is bounded _____ if there exists a b in B such that a _\<_ b for all a in A. And b is an ______ bound for A.

Answer 42

Bounded Above

Question 43

A set A _c_ R is bounded ____ if there exists b in **R** such that b _\<_ _a for all a in A._ And b is an ____ bound for A.

Answer 43

Bounded Below

Question 44

for some b in **R**, b is the ____ bound if for a set A _c_ **R** if: i. ) b is an _____ bound of A ii. ) if c is any ____ bound of A, b _\<_ c

Answer 44

Least Upper Bound

Question 45

for some b in **R**, b is the ____ bound if for a set A _c_ **R** if: i. ) b is an _____ bound of A ii. ) if c is any ____ bound of A, b _\>_ c

Answer 45

Greatest Lower Bound

Question 46

a function f:A to B is one-to-one if a₁≠a₂ and f(a₁)≠f(a₂)

Answer 46

One to One

Question 47

f is onto if given b in B, there exists a in A such that f(a)=b

Answer 47

Onto

Question 48

Given a function f:D to **R** and a subset B _c_ **R**, let _____ be the set of all points from the domain D that get mapped into B \_\_\_\_={x in D: f(x) in B}

Answer 48

Pre-image

Question 49

there exists some a in A satisfying s - ε \< a

Answer 49

Theorem- Assume s in **R** is an upper bound for a set A _c_ **R**. Then s=sup(A) if and only if, for every choice of ε \> 0....

Question 50

there exists some a in A such that i + ε \> a

Answer 50

Theorem- Assume i in **R** is a lower bound for a set A _c_ **R**. Then i = inf(A) if and only if for every choice of ε \> 0....