Axioms and Postulates.pptx

Axioms and Postulates

1. Mathematical System and Axiomatic Structure Undefined Terms, Defined Terms, Theorems, and Postulates/Axioms

2. Axioms and Postulates Euclid, starting from his definitions, assumed certain properties, which were not to be proved. These assumptions are actually ‘obvious universal truths’, in which he divided into two: axioms and postulates.

3. Axioms and Postulates Assumptions that were specific to Geometry are called POSTULATES Common notions or AXIOMS, on the other hand, were assumptions used throughout Mathematics and not specifically linked to Geometry

4. Axioms Things which are equal to the same thing are equal to one another If equals are added to equals, the wholes are equal. If equals are subtracted from equals, the remainders are equals.

5. Axioms Things which coincide with one another are equal to one another. The whole is greater than the part. Things which are double of the same things are equal to one another. Things which are Half of the same things are equal to one another.

6. Postulates A straight line segment can be drawn joining any two points Any straight line can be extended indefinitely in a straight line. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

7. Postulates A straight line segment can be drawn joining any two points Axiom 5.1 Given two distinct points, there is a unique line that passes through them.

8. Postulates All right angles are congruent. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

9. Theorem 5.1 Two distinct lines cannot have more than one point in common.

10. Exercise Which of the following statements are true and which are false? Give reasons for your answers. Only one line can pass through a single point. There are an infinite number of lines which pass through two distinct points. Any straight line can be extended indefinitely in a straight line. If two circles are equal, then their radius are equal.

11. Exercise 2. Give a definition for each of the following terms. a. parallel lines d. radius of a circle b. perpendicular lines e. square c. line segment

12. Exercise 3. Consider two postulates given below: a. Given any two distinct points A and B, there exists a third point C which is in between A and B. b. There exist at least three points that are not on the same line. Are these postulates consistent? Do they follow Euclid’s postulates? Explain.