Mat 1362 Midterm Two

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)