What is the difference between theorems and properties and axioms

An axiom is the most basic, unproved, primordial true proposition that people choose when establishing an axiom system, so an axiom can be considered a kind of nature.

A theorem is a true proposition deduced from an axiom or other theorem in an axiom system, so it can be proved. The theorem can also be considered as a kind of property.

People want to create a theoretical system, as the basis and starting point of the theory, they will artificially stipulate some principles without proof, such as: after two points, there is only one straight line. These principles are universally recognized and self-evident.

Because it is the most primitive, the most basic, and the artificial, the proof of the axiom is meaningless.

Starting from the axioms, some basic commonly used laws called theorems are deduced, which enrich the theoretical system and provide more convenient tools and a higher theoretical platform for the development of theories.

Properties are the understanding of theorems from different perspectives, or the more commonly used laws derived from theorems.

Axioms, theorems, and properties constitute a complete theoretical system, and to overturn this set of theories is to overturn the axioms of a theoretical system that is more in line with the law.

A theorem is a conclusion that starts from a true proposition (an axiom or other theorem that has been proven) and proves to be correct after deductive deduction that is limited by logic, i.e., another true proposition.

Quality. It has characteristics that are different from other things.

Axiom. A true proposition in a deductive system that does not need to be proved as a starting point.

The axiom does not need to be proved to be recognized, and the theorem needs to be proved by the axioms, and the properties are what distinguish the matter itself from other substances.

Axiom. In the previous sections, we learned about the properties of some figures, all of which are true propositions. Some of them.

propositions, such as "two points determine a straight line", "two straight lines are truncated by a third straight line, if.

If the isotopic angle is equal, then these two straight lines are parallel" and so on, and their correctness is what people are doing in the long run.

Summarized in practice, and used as the basis for judging the truth or falsity of other propositions, such true propositions are called.

for axioms. Theorems and proofs.

There are also propositions such as "the opposite top angles are equal", "two straight lines are parallel, and the inner wrong angles are equal".

etc., their correctness is confirmed by reasoning, and such true propositions are called theorems, reasoning processes.

It's called proof. Below, we use proof "" to illustrate what proof is.

Bright. As can be seen from this example, the proof is a step-by-step process based on the question (known).

reasoning, and finally the process of deducing the conclusion (verification) of the correct.

Note that every step of reasoning in the proof must be well-founded and cannot be "taken for granted". These roots.

It can be a known condition, a definition, an axiom, a theorem that has already been learned. In the beginning of learning.

When proving, it is required that the basis be written in parentheses after the first step of reasoning, in which the substitution of the same quantity, the use of the properties of the equation to add, subtract, multiply, and divide and other algebraic operations can be done without reason.

1. Assertive theorem: Yes.

to determine whether the thing under discussion at the source conforms to a theorem of a certain concept (or axiom, mathematically speaking), a theorem is a sufficient condition for satisfying a certain concept (axiom), so the main function of a theorem is to judge.

2. Property theorem: It is a theorem obtained from concepts (axioms). The property theorem can be derived directly from the concept (axioms), and when discussing a concept, it contains all its properties, so the main function of the property theorem is to describe.

There is a difference in the conditions given.

1. The theorem is applicable to judging whether the nature of the thing in question conforms to a certain concept.

2. The property theorem is to deduce concepts based on the given properties.

A decision theorem is that a known parallel or perpendicular is extrapolated to other outcomes, and a property theorem is a conditional extrapolation of parallel or perpendicular results.

A definition is a shorter, more explicit proposition that reveals the nature of a thing as reflected in a concept.

Theorems are propositions that are deduced from definitions and axioms.

An axiom is a proposition that is tacitly true in a system of theorems, and theorems are based on axioms or other true propositions.

theorem).

Illustrate with a line segment as an example.

The definition, the concept of these two are the same, both are shouting at the description of a thing, as far as Zheng Shanhe is saying "what is called a line segment", what is a line segment.

Axiom: This cannot be proved, but it is indeed true, it is a law. For example, the shortest line segment between two points is known, but it cannot be proven.

Theorem: This is proved by other theorems or axioms or definitions, and it differs from axioms in that it can be proved, but it is often true within certain conditions, and it may not be true without conditions.

Properties: Sometimes it is also said to be a property theorem, which can be proven, but because it is more intuitive, as a property, for example, the diameter of a circle is equal.

In general, it can be divided into three categories, definitions and concepts, axioms, properties and theorems.

Difference Between Axiom and Theorem:

Axiom": It is the basic mathematical knowledge that people have summarized in long-term practice, and is used as the basis for judging the truth or falsity of other propositions.

Theorem": The true proposition obtained by the method of reasoning is called "theorem", and the method of this reasoning is also called "proof".

Brief introduction. In mathematics, the word axiom is used in two related but distinct meanings – logical axioms and non-logical axioms. In both senses, axioms are used as starting points for deriving other propositions.

Unlike theorems, an axiom (unless there is redundancy) cannot be deduced from other axioms, otherwise it is not the starting point itself, but some kind of result that can be derived from the starting point—it can simply be reduced to a theorem.

Difference Between Theorem and Axiom: An axiom is a conclusion that cannot be proved but is indeed correct, and is an objective law. A theorem is a correct conclusion that is deduced and proved by an axiom under certain conditions.

In mathematics, a theorem is a proposition that is proved on the basis of an existing proposition, which can be another theorem or a widely accepted statement, such as an axiom. The proof of a mathematical theorem is a process of inference about the proposition of the theorem in a formal system. The proof of a theorem is often interpreted as a verification of its authenticity.

It can be seen that the concept of theorem is basically deductive, which is different from other scientific theories that need to be supported by experimental evidence.

Axioms refer to basic propositions that do not need to be proved after being tested by human beings through long-term and repeated practice based on the self-evident basic facts of human reason. In mathematics, axioms are used as a starting point for deriving other propositions. Unlike theorems, an axiom (unless there is redundancy) cannot be deduced from other axioms, otherwise it is not the starting point itself, but some kind of result that can be derived from the starting point—it can simply be reduced to a theorem.