MATHEMATICS 152, FALL 2008THE MATHEMATICS OF SYMMETRYOutline #1 (Proof)Last modified: September 4, 2008Reading. Biggs, Sections 1.1-6, 3.1-5. Reading Biggs means working someexercises as you go, just for practice. Also be sure to read and think about theexercises in Section 1.2, which have interesting discussions (the ‘discussions’ arenot solutions).In this class we won’t bother with the logical notations such as ¬ and ∧ whichBiggs uses in Chapter 3 (the notation =⇒ for ‘implies’ is nice though). Nor willwe bother with truth tables. However, he uses this language in the text, so youwill have to get used to it there in order to read Chapter 3.1. In proper formal mathematical language, write clearly and present• The definition of a prime number.• A theorem about prime numbers of the form “The number n is not aprime number.” (That is, you pick an n.)• A proof of that theorem.• A theorem about prime numbers of the form “The number n is a primenumber.”• A proof of that theorem.(Biggs 1.6)2. • Explain the terms universal statement and existential statement. Il-lustrate each with three or four examples, some of which are true andsome of which are false.• Explain the term counter-example and give some counter-examples tothe false universal statement(s) you gave in the first part.• Give a statement which could be considered both a universal and anexistential statement.(Biggs 1.4-5)3. Explain the concept of negation. (Don’t bother with Biggs’ notations ¬etc., but you can just write “NOT( statement )”) What is the negation of‘A and B’ ? What is the negation of ‘A or B’ ? What is the negation of ‘Band not A’? Explain why. (Biggs 3.1-2)14. Consider the statement “If I am from mars, then you are from venus.” Isthis true or false? Explain. (Biggs 3.3)5. Explain and give examples of the concepts of ‘contrapositive’, ‘converse’and ‘if and only if’ (don’t bother with Biggs’ notations ¬ etc.) Explainthe relationship between a statement and its contrapositive. Explain therelationship between a statement and its converse. (Biggs 3.4-5)Consider the statement ‘a perfect square which is odd must be the squareof an odd number.’• Put the statement in ‘if - then’ theorem form. (Hint: You can beginwith a hypothesis before the ‘if - then’ part of the theorem if you like.)• Prove the statement by stating its contrapositive and proving that.• State and prove the converse of the statement.• State a theorem using the phrase ‘if and only if’ that combines yourtwo results.6. Give the proof in Exercise 1.6.4 in Biggs and help the class discover whatis wrong with it.7. Give the following Definitions and Theorems (we can assume these theoremsare true):• Definition. A flog is a greeb that has positive appoplactation.• Definition. A gawaxian is a flobbert that has inert feebles.• Theorem. Any ablutareen is either a greeb or a flobbert.• Theorem. Any ablutareen with inert feebles has positive appoplac-tation.Now prove the following theorem.Theorem. Any ablutareen with inert feebles is a flog or a gawaxian.8. Explain what proof by contradiction means. Show that 8 is not a perfectsquare by proof by