Basic concepts in mathematics
Mathematics is divided into two parts: pure mathematics and applied mathematics. The first is concerned with creating problems out of nothing by studying and generalizing properties of something or proving a hypothesis.
The second concerns finding solutions to an existing problem
One of the most important foundations of mathematics is "foundation of mathematics".
It is set theory and a set is defined as a collection of different unordered elements that share a certain property and we denote any set with the letters "capital", how can we represent any set!! There are two ways to represent the set either by listing its elements A={...} or by describing these elements as {A={x∈B:p(a)]
read => elements of set A are all elements of a belonging to domain B so that ":" which fulfills P(a), the P(a) condition
for example:
The set of numbers that are divisible by 4
We can represent it as a group by listing its elements
A={...,-8,-4,0,4,8,...}
or
describing its components
A={a ∈ Z: a=4m , m ∈ Z}
Another example, solutions of the equation "x²=1 ,x ∈ ℝ"
A={1,-1} and the same A={ x ∈ ℝ : x²=1}
A space is defined mathematically as a set, sometimes called a univere, with a structure defined on it. With the fulfillment of certain conditions represented by the axioms axioms
Denotes it (X, structure)
Structure: A set or more associated with mathematical objects such as binary operations, relationships, or subsets
A mathematical object is anything that has properties in mathematics
Line, curve, set, matrix, operations, relations, space itself are all mathematical objects
For example: We took the vector space in liner.
It is a set of elements called "Vectors".
The structure defined on it is an array called a "Field"
and two operations "addition and scalar multiplication"
With the fulfillment of 10 "Vector space axioms" conditions
We can symbolize it (X,+,•)
The importance of the "Field" is that I take the "scalars" from it.
For the record, the vectors in "Vector space" do not necessarily represent just geometric vectors like the ones we took in physics or Calculus 3, they can represent any mathematical object that fulfills the 10 conditions like, for example, a polynomial of degree "n≥1" with real coefficients
The field "Field": a set that fulfills certain conditions "field axioms" and is defined by the operations of addition (a+b) and multiplication (a•b) which we know "0" is the neutral element by addition and "1" is the neutral element by multiplication, from the definition of the additive counterpart (a-) and the multiplicative counterpart (a-¹) for all elements in the set except zero (the inverse condition of the axioms is that a-¹ is the inverse for all elements of the set except zero a≠0)
From this definition, we can consider that the set is defined by 4 operations of addition, subtraction, multiplication and division that we know them
A group: a group for which only a unit process is defined and with the conditions of the group "group axioms" being met.
Note, that the names “conventions” are abstractions of the concepts that we know, and the mathematical term does not necessarily express the things that we understand, for example, like the term “dimension” when we talk about a vector with an “n” dimension, we do not necessarily mean that in reality we have more than the third dimension.
For this reason, we always need to put definitions of mathematical terms in order to be able to imagine the concept that this term expresses
From this point of view, during your study, do not pay much attention to the nomenclature, the most important thing is to realize the characteristics that are addressed by the definition
Mathematics is very good at describing and representing reality. Nothing can be expressed mathematically by the empty set { }
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