MAT 1362 - Final and Combined Flashcards

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.

What is an infinite sequence

A list of integers xj for j element of N.

How do you denote a infinite sequence

(xj) (j=1) to (inf)

What is a finite sequence

A list of integers xj for j element of a finite set.

Finite sequence notation

is typically denoted as (xj) for j=1 to n, where n is the number of elements in the sequence.

What is summation

The process of adding a sequence of numbers together, typically represented with the sigma notation (∑).

What is product

The process of multiplying a sequence of numbers together, often represented with the pi notation (∏).

How to define a set Z

Z := {n EZ : n>=0}

What does := mean

It denotes 'is defined as' in mathematical notation, indicating the definition of a variable or set.

What is the factorial

The factorial is the product of all positive integers up to a given number, denoted by n!, where n is a non-negative integer.

What is the factorial rules (2)

1) 0! = 1

2. n! = n × (n - 1)! for n > 0

What is Theorem 4.10 (Factorial of k!)

k! is divisible by m!(k-m)!

(^k_m)

“K choose M”. It is equal to the number of ways of choosing m objects from a collection of k objects, calculated as k! / (m!(k-m)!).

Coefficients can be obtained from

Pascals Triangle

What is theorem 4.12 (Binomial Theory for integers)

If a, b, EZ and k EZ(>=0), then (a+b)^k= \sum_{m=0}^{k} {k \choose m} a^{k-m} b^m.

Theorem 4.14 (Principle of Induction, second form)

For each k EN, let p(k) be a statement. Then

i) p(1) is true

ii) If p(j) is true for all ints j s.t 1<=j<=n then p(n+1) is true.

Then p(k) is true for all k EN

A ⊆ A

Set containment is reflexive

if A ⊆ B and B ⊆C, then A ⊆ C

Transitivity

∅

Empty Set

AnB

{x: xEA and xEB}

AnB = ∅

if A and B are disjoint sets

Theorem 5.9 (De Morgan’s Laws)

suppose A,B ⊆ X

i) (AnB)^c= A^c ∪ B^c

ii)(AuB)^c= A^c n B^c

Union and intersection are

associative

What is a relation on a set

that describes how elements from one set are related to elements of another set. It can be defined as a subset of the Cartesian product of two sets.

Definition 6.4 → Equivalence Relation, Equivalent Class

A relation that is reflexive, symmetric, and transitive, defining a way to group elements into equivalence classes.

~

is a relation that is reflexive, symmetric, and transitive.

What is AxB

the Cartesian product of sets A and B, consisting of all ordered pairs (a, b) where a is in A and b is in B.

empty set x A =

empty set

A function consists of

i) a set A called the domain

ii) a set B called the co-domain

iii) a ‘rule’ f that ‘assigns’ to each a EA and exactly an f(a) EB

Denotation of an function

f: A → B

What is a partition

A partition of a set is a grouping of its elements into non-empty subsets, such that every element is included in exactly one subset and no two subsets overlap.

What are the rules of Parition

i)p1, p2 E pi p1dnep2 → p1np2 = empty set

ii)Every a E A belongs to some p E pi

What is the absoulte value

The absolute value of a real number is its distance from zero on the number line, regardless of direction. It is denoted as |x|, where x is the number.

Theorem 6.13 → The division Algorithm

suppose n EN. For every m EZ. there exists a unique q,r EZ s.t

m = qn+r and 0<=r<=n-1.

The integer q is called the quotient and r is called tge remainder upon divison of m by n

x=- y mod n

x-y is divisble by n

Set of equivalence classes is denoted by

Z/nZ or Z_n

What are the opperations on Zn

⊕ ⊙

[a] ⊕ [b]

[a+b]

[a] ⊙ [b]

[a*b]

Prop 6.25 (relation of ax 1)

Fix an int n>= 2. Axiom 1.1 → 1.4 hold when Z replaced by Zn

Definition of prime numbers

An int n>= 2 is prime if it is divisble only by ± 1 and ± . If an int n= 2 is not a prime, it is composite. If n = q1 * q2 … qn EZ, then the q1 … qn are called factors of , and rhe expression n=q1 … qk is called a factorization of n

What is prime factorization

The expression of a composite number as a product of its prime factors. For example, the prime factorization of 12 is 2 × 2 × 3.

What is fermats little theorem

If we have an m EZ and p is prime, then m^p = m(mod p) for any integer m.

What is m^p-1 equalvalent to

1 (mod p)

Definition of Axiom

Statement or proposition, accepted as true without proof.

Definition of Set

Collection of things labeled S, obtains elements/members

Binary Operation

Action on a set, they define a function. Typically depend on two or more elements.

Everything is connected to ______ and ______ in a set

addition, multiplication

Axiom 1.1 → Commutativity, associatory and distributivity

1. a+b = b+a (C)

2. (a+b)+c = a+(b+c) (A)

3. a(b+c) = ab + ac (D)

4. ab = ba (C)

5. (ab)c = a(bc) (A)

Axiom 1.2 → Addition Identity

There exists an integer 0 s.t a+0 = afor all integers a.

Axiom 1.3 → Multiplicative

Identity There exists an integer 1 such that a×1 = a for all integers a.

Axiom 1.4 → Additive Inverse

For every integer a, there exists an integer -a such that a + (-a) = 0.

Axiom 1.5 → Cancellation

If a, b, and c are integers and a + b = a + c, then b = c.

Properties of =

1. a = a (Reflexivity)

2. a=b, then b=a (symmetry)

3. a=b, b=c, then a=c (Transitivity)

4. a=b, then a can be replaced by b in any statement or expression

Properties of Does not equal

1. NOT reflexive

2. Symmetric

3. NOT transitive

How do you mark the end of a proof

with a square