1. Statements that judge things are called propositions, correct propositions are called true propositions, and wrong propositions are called false propositions.

2. In the proposition in the form of "if P, then Q", P is called the condition of the proposition and Q is called the conclusion of the proposition.

Second, four propositions

1. For two propositions, if the conditions and conclusions of one proposition are the conclusions and conditions of the other, then these two propositions are called reciprocal propositions, one of which is called the original proposition and the other is called the inverse proposition of the original proposition.

2. For two propositions, if the conditions and conclusions of one proposition are the negation of the conditions and conclusions of the other, then these two propositions are called mutually negative propositions, one of which is called the original proposition and the other is called the negative proposition of the original proposition.

3. For two propositions, if the condition and conclusion of one proposition are the negation of the conclusion and condition of the other proposition, then these two propositions are called mutually negative propositions, one of which is called the original proposition and the other is called the negative proposition of the original proposition.

Third, the relationship between the four propositions.

1, the relationship between four propositions: the original proposition and the inverse proposition are reciprocal, the inverse proposition and the inverse proposition are reciprocal, the inverse proposition and the original proposition are reciprocal, the original proposition and the inverse proposition are reciprocal.

2. The relationship between the truth and falsehood of the four propositions: (1) The two propositions are mutually negative and have the same truth and falsehood. (2) Two propositions are mutually contradictory propositions or mutually contradictory propositions, and their truth is irrelevant.

Four. Sufficient and necessary conditions

1, "If P, then Q" is a true proposition, which is called the derivation of Q from P, and is denoted as p =>q. It is also said that P is a sufficient condition for Q, and Q is a necessary condition for P.

2. "If P, then Q" is a false proposition, which means that Q can't be deduced from P, and it is marked as P ≦ >; Q, and said that p is not a sufficient condition for q, and q is not a necessary condition for p.

Necessary and sufficient conditions for verbs (abbreviation of verb)

If two p =>q and q =>p, just write it as P.

6. Simple logical connectives

(1) and

1. Linking P and Q with the conjunction "and" is called a new proposition, which is recorded as p∧q and pronounced as "P and Q".

2, the proposition p∧q is true or false:

p q p∧q

Really, really.

Right or wrong

Fake, real, fake

False false false false.

(2) or

1. Linking P and Q with the conjunction "or" is called a new proposition, which is recorded as p∨q and pronounced as "P or Q".

2, the proposition p∨q true and false judgment:

p q p∨q

Really, really.

True or false.

true and false

False false false false.

(3) No.

1. If you deny a proposition p, you will get a new proposition, which is recorded as ┐p and pronounced as "non-p".

2, the proposition ┐p's true and false judgment:

┐p

Right or wrong

False truth

Seven. Full name quantifier and existential quantifier

Words such as 1, "to all" and "to any one" are logically called universal quantifiers, which are recorded as "inverted A", and propositions containing universal quantifiers are called universal propositions.

2. for any x in m, p(x) holds, and it is recorded as "inverted a" x ∈ m, p(x).

3. Words such as "You Yi" and "At least One" are called existential quantifiers in logic, marked as "anti-E", and propositions containing existential quantifiers are called special propositions.

4. There is an X in M, which makes p(x) hold, and it is recorded as "anti-E" X ∈ M, p(x).

Eight, the negation of the proposition containing quantifiers

1. The negation of the full-name proposition P: "inverted A" x ∈ m, p(x) ┐p is: "anti-e" x ∈ m, ┐p(x).

2. The negation of the special proposition P: "anti-E" x ∈ m, p(x) ┐p containing quantifiers is: "inverted A" x ∈ m, ┐p(x).

Nine, the "elements of geometry" proposition (especially)

Especially the proven propositions in Euclid's Elements of Geometry, namely the following 48 propositions:

1. Make an equilateral triangle on a known finite line.

2. The line segment composed of known points (as endpoints) is equal to the known line segment.

3. Given two unequal line segments, try to cut one line segment from the top to make it equal to the other.

4. If two triangles have two sides equal to two sides respectively, and the angles contained in these two equilateral line segments are equal, then their bottoms are equal to the bottoms, all triangles are equal to triangles, and the remaining angles are equal to the remaining angles, that is, the angles opposite to these sides.

5. In an isosceles triangle, the two base angles are equal; And if the waist extends downward, the two angles below the bottom are equal.

6. If two angles in a triangle are equal, the opposite sides of the two angles are also equal.

7. If two line segments intersect at one point on a known line segment (from its two endpoints), it is impossible to make the other two line segments intersect at another point on the same side of the line segment (from its two endpoints), so that the two line segments made are equal to the previous two line segments respectively. In other words, the line segments from each intersection point to the same endpoint are equal.

8. If one of two triangles has two equal sides, and the base of one of them is equal to the base of the other, the angles sandwiched between the equal sides are also equal.

9. Divide the known straight angle equally.

10. Divide the known finite straight line equally.

1 1. A straight line drawn from a known point on a known straight line is at right angles to the known straight line.

12. A known point outside a known infinite straight line is regarded as the perpendicular of the straight line.

13. The adjacent angle formed by the intersection of a straight line and another straight line is either two right angles or equal to the sum of two right angles.

14. If two straight lines intersect a point on any straight line and are not on the same side of this straight line, and the sum of adjacent angles with these two straight lines is equal to two right angles, then these two straight lines are on the same straight line.

15. If two straight lines intersect, their antipodes are equal.

16. In any triangle, if one side is extended, the outer angle is greater than any inner angle.

17. In any triangle, the sum of any two angles is less than two right angles.

18. In any triangle, the long side faces the big corner.

19. In any triangle, the big angle is opposite to the big side.

20. In any triangle, the sum of any two sides is greater than the third side.

2 1. If the two endpoints on one side of a triangle are two intersecting line segments within the triangle, the sum of the line segments from the intersection to the two endpoints is less than the sum of the line segments on the other two sides of the triangle. However, the included angle is greater than the vertex angle of the triangle.

22. Try to form a triangle by three line segments equal to three known line segments respectively: in such three known line segments, the sum of any two line segments must be greater than the other line segment.

23. Make a straight line angle equal to the known straight line angle on the known straight line and a point on it.

24. If two sides of a triangle are equal to two sides of another triangle, and the included angle of one triangle is greater than that of the other triangle, the opposite side with larger included angle is also larger.

25. If in two triangles, two sides of one triangle are equal to two sides of the other triangle, then the angle opposite to the larger third side is also larger.

26. If in two triangles, two angles of one are equal to two angles of the other, and one side is equal to one side of the other. That is to say, this side is either an equiangular clip or an equiangular opposite side. Then their other sides are equal to other sides, and other angles are equal to other angles.

27. If the staggered angle formed by the intersection of a straight line and two straight lines is equal, the two straight lines are parallel to each other.

28. If the congruence angle formed by the intersection of a straight line and two straight lines is equal, or the sum of the internal angles on the same side is equal to two right angles, then the two straight lines are parallel to each other.

29. When a straight line intersects with two parallel straight lines, the internal angle is equal, the complementary angle is equal, and the sum of the internal angles on the same side is equal to two right angles.

30. If some lines are parallel to the same line, they are also parallel to each other.

3 1. Make a straight line parallel to the known straight line after passing the known point.

32. In any triangle, if an edge is extended, the external angle is equal to the sum of two internal angles, and the sum of three internal angles of the triangle is equal to two right angles.

33. Connect (respectively) equal and parallel line segments in the same direction, which are themselves equal and parallel.

34. In a parallelogram patch, the opposite sides are equal, the diagonal lines are equal, and the diagonal lines bisect the patch.

35. Parallelograms on the same base and between the same two parallel lines are equal to each other.

36. An equilateral parallelogram and a parallelogram between two identical parallel lines are equal to each other.

37. The triangle on the same base is equal to the triangle between the same two parallel lines.

38. The equilateral triangle and the triangle between the same two parallel lines are equal to each other.

39. An equilateral triangle with the same base and the same side must be between the same two parallel lines.

40. An equal triangle with equal base and on the same side of the base is also between the same two parallel lines.

4 1. If a parallelogram and a triangle are on the same base and between two parallel lines, then the parallelogram is twice as big as the triangle.

42. Use a known right angle as a parallelogram to make it equal to a known triangle.

43. In any parallelogram, the complement of the parallelogram on both sides of the diagonal is equal.

44. Make a parallelogram with known line segments and known straight line angles, so that it is equal to a known triangle.

45. Make a parallelogram with a known straight line angle to make it equal to a known straight line.

46. Draw a square on the known line segment.

47. In a right triangle, the square of the opposite side of the right angle is equal to the sum of the squares of both sides of the right angle.

48. If in a triangle, the square of one side is equal to the sum of the squares of the other two sides of the triangle, then the angle between the back two sides is a right angle.