Start Learning Mathematics

Introduction

In my goal to learn about GF(256) I have landed on this course. I am expecting to know most of this but maybe the terminology might be not known.

Logical Statements and Operations

Logical Statements can either be True or False
(a) Mars is a planet (True logical Statement)
(b) Moon is a plant (False logical Statement)
(c) 1+1 = 2 (True logical Statement)
(d) x+1 = 1 (Not logical Statement as we do not knows a value of x)
Logical Operations are
Negation For a logical statement A, ¬A denotes the negation. (¬ This is the NOT symbol)
Conjunction For two logical statements A,B, A∧B denotes conjunction (∧ This is the AND symbol)
Disjunction For two logical statements A,B, A∨B denotes and disjunction (∨ This is the OR symbol)
So they went of to write truth table the statement ¬A∨A
So the truth table is always written with the variable first

  A | ¬A | ¬A∨A
  T |  F | T
  F |  T | T

So in this case all ¬A∨A is a tautology because the answer is always true

Two logical statements are called logically equivalent if the truth tables are the same
So looking at ¬(A∨B) and (¬A)∧(¬B) we get

 A  | B  || A∨B | ¬A | ¬B | ¬(A∨B) | (¬A)∧(¬B) 
------------------------------------------------
 T  | T  ||  T  |  F |  F |   F     |    F
 T  | F  ||  T  |  F |  T |   F     |    F
 F  | T  ||  T  |  T |  F |   F     |    F
 F  | F  ||  F  |  T |  T |   T     |    T

This is denoted by this symbol $\iff$ so we can write $\mathrm{\neg }(A\vee B)\iff (\mathrm{\neg }A)\wedge (\mathrm{\neg }B)$

Sets

A Set is a collection of distinct objects into a whole
Such an Object x inside set M is called an element of M. Which is denoted by $x\in M$. If x is not a member we cross out the e $x\notin M$
A set can be defined by giving all its elements: $A:=\{2,5,6\}$
Empty set is denoted my $\varnothing :=\{\}$
Natural Numbers: $\mathbb{\mathbb{N}}=\{1,2,3,\dots \}$
Natural Numbers (including zero): ${\mathbb{\mathbb{N}}}_0=\{0,1,2,3,\dots \}$
Integers: $\mathbb{\mathbb{Z}}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$
Rational Numbers: $\mathbb{\mathbb{Q}}$
Real Numbers: $\mathbb{ℝ}$
Complex Numbers: $\mathbb{ℂ}$

Predicates

The formal definition of a predicate is
    A predicate is a function whose output is a truth value
Examples
    $D(a,b):=(\mathrm{\exists }k\in \mathbb{\mathbb{Z}}\mid b=ak)$
    $A:=\{x\in \mathbb{\mathbb{N}}\mid x>10\}$

I am more used to them when using C#

numbers.Any(x => x > 100);  // predicate
numbers.All(x => x >= 0);   // predicate

Forming New Sets

We can form new sets by using predicates. For example
    $A=\{x\in \mathbb{\mathbb{N}}|Xisanevennumber\}$ This should be read as The set of all x in N that satisfy X is an even number