MAT 1362 - Final Stretch

What axioms hold in the reals

The axioms that hold in the reals include the properties of closure, associativity, commutativity, distributivity, identity elements, and the existence of inverses for addition and multiplication, as well as the completeness property.

Ax 7.5 - Multiplicatove Inverse

For every non-zero real number, there exists a multiplicative inverse such that their product is 1.

What is a priori in math

The same noation in a different universe

Ax 7.13

For every R_>0 subset R with the following properties

i) if xy ER_>0, then x*y ER

ii) if xy ER_>0, then x+y ER

iii) 0 !E R_>0

iv) For every x ER, we have x ER_>0 or x=0 or -xER_>0

What are the elements of R_>0 called

Positive real numbers

Defintion → Ordering in R

For x,y ER, we write x<y or y>x iff y-x ER_>0.

How do we convert a prop to the reals

Add “for R” to the defintion

Does AX 1.5 imply AX 1.7

NO

Does Ax 1.7 imply 1.5

YES

What does abounded above mean

A set is said to be bounded above if there exists a real number that is greater than or equal to every element in the set.

What does bounded below mean

A set is said to be bounded below if there exists a real number that is less than or equal to every element in the set.

What is the Supremum

The supremum of a set is the least upper bound, which is the smallest real number that is greater than or equal to all elements in the set.

What is the Infimum

The greatest lower bound of a set, which is the largest real number that is less than or equal to every element in the set.

what is sup (A)

Supremum

What is inf(A)

Infimum of set A

What set has no upper bound and a lower bound of 0

is the set of all non-negative real numbers.

Is it popssible not to have a sup or inf

yes

Does the sup/inf need to be in the set

No, the supremum or infimum does not need to be an element of the set.

Steps to solve for a Sup

involves identifying the least upper bound of a set by evaluating its elements and determining the smallest number that is greater than or equal to all members of the set.

Steps to solve for an Inf

involves identifying the greatest lower bound of a set by evaluating its elements and determining the largest number that is less than or equal to all members of the set.

Def 7.31 → Max, min

Suppose A is a subset of R with A ≠ 0

i) An element bEA is the max element of A if for alll aEA, A<=B. If this is the case, we will write b = min(A)

ii)An element b ∈ A is the min element of A if for all a ∈ A, b ≤ a. If this is the case, we will write b = max(A).

Ax 7.35 → Completeness Axiom

Every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) in the real numbers.

If we have x=y, then

[x,x]={0}

If we have xy, then

[x,y]=(x,y)={0}

Def 8.1 → Injection

A function f: A → B is called an injection if for every pair of distinct elements a1, a2 in A, f(a1) ≠ f(a2).

Def 8.4 → Surjection

A function f: A → B is called a surjection if for every element b in B, there exists at least one element a in A such that f(a) = b.

Def 8.6 → Bijection

A function f: A → B is called a bijection if it is both an injection and a surjection, meaning it pairs every element in A uniquely with every element in B without any omissions.

What is composite

A function formed by combining two functions, where the output of one function becomes the input of another. If f and g are functions, the composite function is denoted as (g.f)(x) = g(f(x)).

g.f: A→ B

(g.f)(a)=(g(f(a))

If f and g are both injective

functions, then their composite function g.f is also injective.

If f and g are both surjective

functions, then their composite function g.f is also surjective.

If f and g are both bijective

functions, then their composite function g.f is also bijective.

Def 8.13 → Inverse Function

If a function is bijective, then it has an inverse function that reverses the mappings of the original function.

What is a left inverse

A function ( g ) is a left inverse of a function ( f ) if ( g(f(x)) = x ) for all ( x ) in the domain of ( f ).

What is the right inverse

A function ( g ) is a right inverse of a function ( f ) if ( f(g(y)) = y ) for all ( y ) in the codomain of ( g ).

What is the two sided inverse (inverse)

A function ( g ) is a two-sided inverse of a function ( f ) if it satisfies both conditions: ( g(f(x)) = x ) for all ( x ) in the domain of ( f ) and ( f(g(y)) = y ) for all ( y ) in the codomain of ( g ).

The function f is injective iff

its left inverse exists.

The function f is surjective iff

its right inverse exists.

The function f is bijective iff

it has both a left and a right inverse.

To be a function what must the set have

a unique output for each input.

If a function is bijective, then it’s _____ is unique

inverse

Def 8.20 → Embedding

Embedding Z in R. We define a function e: Z → R as follows:

i) we define e(0z) = 0R and assuming e(n),for n EZ>0, is defined, we define e(n+Z) = e(n) + 1R.

ii) For k EZ with k<0, we define: e(k) = -e(-k)

What is are the rules for abs

If x \geq 0, then |x| = x; if x < 0, then |x| = -x.

Definiton Distance

For x,y ER, we define the distance between x and y to |x-y|. This matches our intuition of thinking of real nums as points on the real line

Def 9.11 → Limit of a sequence

Let (xk)inf to k=1 be a sequence in R and L ER. We say that (xk)inf to k=1. Converges to L if for all ε >0, there exists N element of Naturals such that for all n>= N |xn-L| <ε. When (xk)inf to k=1 converges if thereexits some L s.t the sequence convereges to L. If no such L exists, then we say the sequence divereges. Thus, the sequence (xk) diverges if for all ER there exists ε>0 s.t for all nEN there exsists n such that |xn - L| ≥ ε.