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M4005 Geometry |
Study Guide for Test February 22, 2006 |
The test will include a selection from the following problems. (6, 7 and 10 will not be on this test.) (Hints and solutions appear in red.)
1) What is a mathematical definition? What is a primitive term? Why does every mathematical theory have primitive terms? Give an example of a primitive term in Euclidean geometry. Point, line. Give an example of a defined term. Parallel lines.
2) What is a postulate? What is a proposition (or theorem)? Why must a deductive system have postulates? State the five Euclidean postulates. State Playfair's Postulate.
The following passage from A. Tarski, Introduction to Logic and to the Methodology of the Deductive Science, Oxford University Press, 1941, Chapter 5 contains a very concise answer to the questions, "What is a mathematical definition? What is a primitive term? Why does every mathematical theory have primitive terms? What is a postulate? What is a proposition (or theorem)? Why must a deductive system have postulates?"
Fundamental constituents of a deductive theory-primitive and defined terms, axioms and theorems
We shall now attempt an exposition of the fundamental principles that are to be applied in the construction of logic and mathematics. The detailed analysis and critical evaluation of these principles are tasks of a special discipline, called the METHODOLOGY OF DEDUCTIVE SCIENCES or the METHODOLOGY OF MATHEMATICS. For anyone who intends to study or advance some science it is undoubtedly important to be conscious of the method which is employed in the construction of that science; and we shall see that, in the case of mathematics, the knowledge of that method is of particularly far-reaching importance, for lacking such knowledge it is impossible to comprehend the nature of mathematics.
The principles with which we shall get acquainted serve the purpose of securing for the knowledge acquired in logic and mathematics the highest possible degree of clarity and certainty. From this point of view a method of procedure would be ideal, if it permitted us to explain the meaning of every expression occurring in this science and to justify each of its assertions. It is easy to see that this ideal can never be realized. In fact, when one tries to explain the meaning of an expression, one uses, of necessity, other expressions; and in order to explain, in turn, the meaning of these expressions, without entering into a vicious circle, one has to resort to further expressions again, and so on. We thus have the beginning of a process which can never be brought to an end, a process which, figuratively speaking, may be characterized as an INFINITE REGRESS--a regressus in infinitum. The situation is quite analogous as far as the justification of the asserted statements of the science is concerned; for, in order to establish the validity of a statement, it is necessary to refer back to other statements, and (if no vicious circle is to occur) this leads again to an infinite regress.
By way of a compromise between that unattainable ideal and the realizable possibilities, certain principles concerning the construction of mathematical disciplines have emerged that may be described as follows.
When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable; the expressions of this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings. At the same time we adopt the principle: not to employ any of the other expressions of the discipline under consideration, unless its meaning has first been determined with the help of primitive terms and of such expressions of the discipline whose meanings have clearly been explained previously. The sentence determining the meaning of a term in this way is called a DEFINITION, and the expressions themselves whose meanings have thereby been determined are accordingly known as DEFINED TERMS.
We proceed similarly with respect to the asserted statements of the discipline under consideration. Some of these statements which to us have the appearance of evidence are chosen as the so-called PRIMITIVE STATEMENTS or AXIOMS (also often referred to as POSTULATES...); we accept them as true without in any way establishing their validity. On the other hand, we agree to accept any other statement as true only if we have succeeded in establishing its validity, and to use, while doing so, nothing but axioms, definitions and such statements of the discipline the validity of which has been established previously. As is well known, statements established in this way are called PROVED STATEMENTS or THEOREMS, and the process of establishing them is called a PROOF. More generally, if within logic or mathematics we establish one statement on the basis of others, we refer to this process as a DERIVATION or DEDUCTION, and the statement established in this way is said to be DERIVED or DEDUCED from the other statements or to be their CONSEQUENCE.
3) Euclid's Proposition 17 asserts that the sum of any two angles in a triangle is less than two right angles. Use this to prove that two lines, both perpendicular to a third line are parallel. Does this argument use the Parallel Postulate?
Assertion: Two lines, both perpendicular to a third line are parallel.
Proof. Suppose lines l and m are perpendicular to line k and let A and B be the intersections of l and m with k. Now consider what it would mean if l and m had a point in common, C, say. Then A, B, and C would be the vertices of a triangle. Proposition 17 would imply that the angles of this triangle at A and B would be LESS than 2 right angles. But we assumed at the beginning that these angles are both right. Therefore, the point C cannot exist. In other words, lines l and m can have no point in common. Hence they are parallel.
This proof does not use the Parallel Postulate, but only the definition of parallel lines. The Parallel Postulate is not needed to prove the existence of parallels.
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Here is a more detailed proof:
(Remark. The hypothesis is that we have two lines both perpendicular to a third. The conclusion is that these lines are parallel. The proof should lead from the hypothesis to the conclusion.)
Proof. Suppose lines l and m are perpendicular to line k and let A and B be the intersections of l and m with k (ekthesis). We wish to show that l and m have no point in common, and hence that they are parallel (diorismos). Consider what it would mean if l and m had a point in common, C, say. Then A, B, and C would be the vertices of a triangle. Proposition 17 would imply that the angles of this triangle at A and B would be LESS than 2 right angles. But our hypothesis is that the angles at A and B are both right. Therefore, under our hypothesis the point C cannot exist. In other words, if our hypothesis is satisfied, then lines l and m can have no point in common. But this is the definition of parallel. Thus, two lines that are both perpendicular to a third are parallel to one another.
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Rather than giving a proof by contradiction (as above), one can prove the contrapositive.
Assertion: Two lines, both perpendicular to a third line are parallel.
Proof. We show that if two (different) lines are NOT parallel, then they CANNOT both be perpendicular to a third line. This is the contrapositive of the assertion, so it is equivalent to the assertion. Suppose two lines l and m are not parallel and a third line k meets l at A and m at B. Since l and m are not parallel, they meet at a point, call it C. If A = B = C, then the angle between l and k and the angle between m and k must be different (since l and m are different lines), so these angles cannot both be right. If A, B and C are different points, then they form the vertices of a triangle. By Proposition 17, the angles at A and B are not both right. So k is not perpendicular to both l and m.
4) Give the definitions of: a) congruence between two labeled triangles, b) similarity between two labeled triangles, c) congruence of two configurations of points in the plane, d) similarity of two configurations of points in the plane. (See January 30 class.)
5) State the SAS, SSS, ASA and AAS criteria for triangle congruence. (No proof is demanded.)
6) Descibe the following constructions: a) bisect an angle, b) perpendicular bisector of a segment, 3) perpendicular to a line through a point not on that line, 4) parallel to a line through a point not on that line.
7) Supply proofs of the constructions in (6).
8) a) Prove that the diagonal of a parallelogram cuts it into two congruent triangles.
Hint: You will use the fact that if parallel lines are cut by a transversal, opposite inetrior angles are congruent. In the proof, you suppose you have a parallelogram. Draw a diagonal to form two triangles with the diagonal as base. Set up a correspondence between the traingles, with base corresponding to base. Use the fact about transversals to get the conditions for ASA.
b) Let ABC be a triangle and let M be the midpoint of AB. Let m be the line through M parallel to BC and let l be the line through C parallel to AB. Let D be the point where l and m meet and let E be the point where m crosses AC. Prove that ECD is congruent to EAM. (See Activity 3 on the February 15 class page.)
Hint: Use the fact about transversals.
9) State and prove Euclid Proposition 35, Book I.
10) State and prove Thales Theorem.