What are the axioms of mathematics and what are the basic facts?

Axiom: Quantities equal to the same quantity are equal to each other. Equal plus equal amount, its sum is equal. The same amount is subtracted by the same amount, and its difference is equal.

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.

By reliable arguments (syllogisms.

Rules of reasoning) The logical deductive method of leading from premises (prior knowledge) to conclusions (new knowledge) was developed by the ancient Greeks and has become a central principle of modern mathematics. Nothing can be deduced except tautology, if nothing is assumed.

An axiom is the derivation of a particular set of basic assumptions of deductive knowledge. The axioms are self-evident, and all other assertions (or theorems if we are talking about mathematics) must be proved with the help of these basic assumptions.

However, the interpretation of mathematical knowledge has been different since ancient times, and ultimately the word "axioms" is as important to mathematicians today as it is to Aristotle.

It is also slightly different from the meaning in Euclid's eyes.

The basic mathematical facts of nature: constant, constant e, equal angles of parallel lines at the same angle, equal angles of internal error, internal angles and radians, Pythagorean theorem a b c, linear space vector synthesis satisfies the parallelogram axioms, complex plane unit imaginary number orthogonal relationship with unit real number (i 丄1), quaternion corresponding to four-dimensional orthogonal space ( i 丄 j 丄 k 丄 1 ).

The axioms of mathematics are:

1. There is only one straight line after two points.

2. The shortest line segment between two points.

3. The complementary angles of the same angle or equal angle are equal.

4. The co-angle of the same angle or equal angle is equal.

5. There is only one and only one straight line perpendicular to the known straight line.

6. Among all the line segments connected by the points outside the line and each point on the line, the perpendicular line segment is the shortest.

7. The axiom of parallelism passes a point outside the straight line, and there is only one straight line parallel to the straight line.

8. If both lines are parallel to the third line, the two lines are also parallel to each other.

1. Two points to determine a straight line.

2. The shortest line segment between two points.

3. Within the same flat Tong Zhao surface, there is only one straight line perpendicular to the known straight line.

4. The same position or carrying angle is equal, and the two straight lines are parallel.

5. There is only one straight line parallel to the straight line.

6. Two triangles with equal angles on both sides and their angles are congruent.

7. Two triangles with equal corners and their edges are congruent.

8. Two triangles with equal sides are congruent.

Proof: a b a

a∩b＜b（a∩b）^c＞a^c

a∩b）^c>b^c

a∩b）^c＞a^c∪b^c……※

The same can be argued, (a b) c a c b c

Substitute a c into a and b c into b so that there is.

a c b c) c (a c) c (b c) c=a b on both sides, get.

a^c∪b^c＞（a∩b）^c

i.e. (a b) c a c b c

Combined with the equation, it can be obtained, :(a b) c = a c b c mathematical set is a fundamental concept mathematically. A basic concept is a concept that cannot be defined by other concepts, nor is it a concept that cannot be defined by other concepts.

The concept of a set can be developed in an intuitive, axiomatic way"Definitions"。

Set (abbreviated set) is a basic concept in mathematics and is the object of study of set theory, which was not created until the 19th century. In the simplest terms, it is defined in the most primitive set theory, naïve set theory, and a set is"A bunch of stuff"。collection"stuff", called an element.

If x is an element of the set a, it is denoted as x a. A set is a collection of certain distinguishable objects in people's intuition or thinking that merge together to form a whole (or monomer), and this whole is a set. Those objects that make up a set are called elements (or simply metas) of the set.

Modern mathematics is still used"Axiom"to prescribe the collection. The most basic axioms are examples: axioms of extension:

For any set s1 and s2, s1=s2 if and only if for any object a, if a s1, then a s2; If a s2, then a s1. There is an axiom of disorder for sets: for arbitrary objects A and B, there is a set S, such that S has exactly two elements, one for object A and one for object B.

By the axioms of extension, the set of disordered pairs composed of them is unique and denoted as. Since a and b are any two objects, they may or may not be equal. When a=b, , can be denoted as or, and is called a set of units.

An empty set existentially axioms: there exists a set, which does not have any elements.

The nine axioms of mathematics in the difference of elementary numbers are two points and there is only one straight line. The line segment between two points is the shortest, the complementary angles of the same angle or equal angles are equal, the co-angles of the same angle or equal angles are equal, there is only one and only one straight line perpendicular to the known straight line at one point, and the perpendicular line segment is the shortest among all the line segments connected by the point outside the line and the points on the straight line. The origin of the nine axioms of junior high school mathematics The axiom of parallelism passes through a point outside the straight line, and there is only one straight line parallel to the straight line, if the two straight lines are parallel to the third straight line, these two straight lines are also parallel to each other, and the two straight lines are parallel to each other with equal isotopic angles, and the axiom is based on the self-evident basic facts of human reason, and the basic proposition that does not need to be proved after the good test of human beings for a long time and repeated practice.

The axiom is the basic mathematical knowledge that people summarize in long-term practice and use it as the basis for judging the truth or falsehood of other propositions, the true proposition obtained by the method of reasoning is called the theorem, the method of this reasoning is also called proof, the theorem is a statement that is proved to be true by the limitation of logic, and the law is a form of expression of the facts of the guest and friend, and the conclusion is summarized through a large number of specific objective facts.

1. The two straight lines are truncated by the third straight line, if the isotopic angle is equal, then the two straight lines are parallel; 2. The two parallel lines are truncated by the third straight line, and the isotopic angles are equal; 3. Two triangles with equal sides and angles are congruent; (SAS 4, the angle and its intermediates correspond to two equal triangle congruence; (asa) 5, three sides correspond to two equal triangles congruence; (sss) 6. The corresponding sides of congruent triangles are equal and the corresponding angles are equal. 7. Line segment axiom: between two points, the line segment is the shortest.

8. Straight line axiom: There is only one straight line after two points. 9. Axiom of Parallelism:

10. Perpendicular properties: After Zheng Cha carefully passed a point outside the straight line or at a point on the straight line, there is only one and only one straight line perpendicular to the known straight line.

In Euclid's "Geometric Primitive", Euclid gave 23 definitions, 5 postulators, and 5 axioms at the beginning. In fact, the commune he spoke of was what we later called the axioms, and his axioms were some methods used for calculations and proofs (e.g., axiom 1: equal to the equal quantity of the same quantity, axiom 5:).

The whole is greater than the part, etc.) The five axioms he gave were very closely related to geometry, which were later the axioms in our textbooks. They are: Public Premise 1:

A straight line can be drawn from any point to any other point Public Hypothesis 2: A finite line segment can continue to be extended Public Hypothesis 3: A circle can be drawn with any point as the center and any distance Public Hypothesis 4:

All right angles are equal to each other Public Hypothesis 5: A straight line in the same plane intersects with two straight lines outside the other plane, if the sum of the two inner angles on one side is less than the sum of the two straight bridge angles, then the two straight lines intersect on this side after being extended indefinitely.

5 cardioms.

He Ming: Vertical rent a b a

a∩b＜b（a∩b）^c＞a^c

a∩b）^c>b^c

a∩b）^c＞a^c∪b^c……※

The same can be argued, (a b) c a c b c

Substitute a c into a and b c into b so that there is.

a c b c) c (a c) c (b c) c=a b on both sides, get.

a^c∪b^c＞（a∩b）^c

i.e. the Troublesome Slim Pie (a b) c a c b c

The combination formula yields, :(a b) c = a c b c