Question

using these axioms prove proof number 5

1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet, they have exactly one point in common Theorem 2: If h, and I are two parallel lines and if linem intersects I (but m l ), then intersects I also. Theorem 3: If each of two lines is parallel to a third line, then the two lines are either parallel or identical Affine Plane proofs 1) Every point is contained in at least two lines. 2) There does not exist a point contained in all the lines. 3) Let / and be intersecting lines. Prove that there is a point not on either line. 4) If x is any point, there exists at least one line not containing x. 5) Every line contains at least one point. 6) Every line contains at least two points. 7) If there exists a line containing exactlyn points, then every line contains exactlyn points 8) If there exists a line containing exactlyn points, then exactly point. lines pass through every 9) If there exists a line containing exactly n points, then there are exactly or points in all. 10) If there exists a line containing exactlyn points, then there are exactly tn lines in all 11) If there exists a line containing exactly given line. points, then there are exactly n. 1 lines parallel to a

Answer 1

E We want to prove every line contains atleast one point. Asciom-2 Says Giren any telco distinct points there is exactly one line incident with both points". . If there is line then this line drauen from two distinct points. Therefore this two I line contains atleast one point o or suppose the line does not contain any point but the line is drawn from two distinct points. so this two points the lie on that line: a contradiction to fact that line does not contain ant point o ___