Test 1

Question 1

Form of an even natural number

Answer 1

2k

Question 2

Form of an odd natural number

Answer 2

2k-1

Question 3

Every non-empty subset of N has a smallest element

Answer 3

Well-Ordering Principle

Question 4

If an integer is odd, then the square of that integer is

Answer 4

odd

Question 5

If n^2 is even, then n is

Answer 5

even

Question 6

Define the rational numbers

Answer 6

Q = {m/n: m,n ∈ Z, n=/=0}

Question 7

There exists no P∈Q s.t. p^2 = 2

Answer 7

True

Question 8

What is the largest element of A = {p∈Q, p>0: p^2<2}

Answer 8

There is no largest element

Question 9

What is the smallest element of B = {p∈Q, p>0: p^2>2}

Answer 9

There is no smallest element

Question 10

Properties of an ordered set

Answer 10

1) One of the following is true: x<y, x=y, y<x (Trichotomy)
2) If x<y and y<z, then x<z

Question 11

Definition of an ordered set

Answer 11

A set in which an order is defined.

Question 12

Definiton of Bounded above

Answer 12

If there exists a B∈S s.t. for any x∈E, x<=B, E is bounded above by B

Question 13

Definition of bounded below

Answer 13

If there exists an A∈S s.t. for any x∈E A<=x then E is bounded below

Question 14

Properties of a supremum (least upper bound)

Answer 14

1) b is an upper bound of E
2) If a<b then a is not an upper bound of E.

Question 15

Properties of infimum (greatest lower bound)

Answer 15

1) a is a lower bound of E
2) if a < b, then b is not a lower bound of E

Question 16

Def of least upper bound property

Answer 16

An ordered set S is said to have the LUB prop if for all E subset S s.t. E=/=0 and E is bound above, then sup(E) exists in S.

Question 17

Def of Field

Answer 17

A set equipped with two opperations, addition and multiplication

Question 18

Addition Axioms

Answer 18

A1) If x,y∈F then x+y∈F
A2) x+y = y+x for all x,y∈F
A3) (x+y)+z = x+(y+z) for all x,y,z∈F
A4) There exists a 0∈F s.t. 0+x=x for all x∈F
A5) There exists a -x∈F s.t. x+(-x)=0 for all x∈F

Question 19

Multiplication Axioms

Answer 19

M1) If x,y∈F then xy∈F
M2) xy = yx for all x,y∈F
M3) (xy)z = x(yz) for all x,y,z∈F
M4) There exists a 1∈F s.t. 1=/=0 and 1x=x for all x∈F
M5) For all x∈F/{0} there exists an x^-1 s.t. x*x^-1 =1

Question 20

Distribution Axiom

Answer 20

x(y+z) = xy+x*z for all x,y,z∈F

Question 21

Properties of Ordered Field

Answer 21

A field which has the following properties:
1)If x<y then x+z<y+z
2) If x,y>0 then 0<x*y

Question 22

Define absolute value of x

Answer 22

x, x>=0
-x, x<0

Question 23

Triangle Inequality

Answer 23

|x+y| <= |x|+|y|

Question 24

Reverse Triangle Inequality

Answer 24

||x|-|y|| <= |x-y|

Question 25

The Archemedian Principle

Answer 25

If x,y∈R and x>0 then there exists n∈N s.t. n*x>y. Hence, for all x∈R there exists an n∈N x

Question 26

Density of the rational numbers

Answer 26

For all x=/=y∈R, there exists a q∈Q s.t. x

Question 27

Def of an interval

Answer 27

Let J be a subset of R. For all x,y∈J with x<=y and for all z∈R s.t. x<=z<=y, z∈J.

Question 28

Def of function

Answer 28

A rule that assigns to each element x∈A excatly one element in set B

Question 29

Def of surjective (onto)

Answer 29

f:A->B is onto if for each b∈B there exists an a∈A with f(a)=b

Question 30

Def of injective (1-1)

Answer 30

f:A->B is 1-1 if for a,b∈A with a=/=b then f(a)=/=f(b)

Question 31

Def of bijective

Answer 31

f:A->B is both injective and surjective.

Question 32

Def of same cardinality (A~B or |A|=|B|)

Answer 32

If there exists a bijection from A into B.

Question 33

Def of relation

Answer 33

R:A->B is a set whose elements are all ordered pairs (x,y) where x∈A and y∈B.

Question 34

Properties of equivalence relations

Answer 34

1) Reflexive 2) Symmetric 3) Transitive

Question 35

Def of finite

Answer 35

If |A|=|Nn| for some n∈N or A={0}

Question 36

Def of infinite

Answer 36

if A is not finite

Question 37

Definite of countable

Answer 37

If |A| =|N|

Question 38

Def of uncountable

Answer 38

If A is neither finite nor countable

Question 39

Def of at most countable

Answer 39

if A is either finite or countable

Question 40

Def of a sequence

Answer 40

A function f:N->A s.t. for each n∈N denote xn=f(n)

Question 41

Def of enumeration

Answer 41

A bijection f:N->A written A={xn: n=1,2,3,...}

Question 42

When does a sequence converge to a∈R

Answer 42

For all ε>0 there exists an N∈N s.t. for all n>=N |xn-a|<ε

Question 43

When does a sequence diverge

Answer 43

If it does not converge

Question 44

When is a sequence bounded

Answer 44

if there exists an M>0 s.t. |xn|<=M for all n∈N

Question 45

Squeeze Thm

Answer 45

{xn}{yn}{zn} are subsets of R with xn->a, yn->a as n->infinity If xn<=zn<=yn for all n∈N then zn->a as n->infinity

Question 46

Increasing sequence

Answer 46

xn+1>=xn for all n∈N

Question 47

strictly increasing sequence

Answer 47

xn+1 > xn for all n∈N

Question 48

decreasing sequence

Answer 48

xn+1<=xn for all n∈N

Question 49

strictly decreasing sequence

Answer 49

xn+1

Question 50

monotone sequence

Answer 50

if {xn} is either increasing or decreasing

Question 51

If {xn} is monotonic and converges then

Answer 51

{xn} is bounded

MAT315 (4 decks)

• Test 1
• Test 2
• Test 3 (Not all just the new ones)
• Final (condensed)