• Explain what is a theorem (proposition) and what is a proof (do not just cite a definition from the dictionary). Be sure to include terms like definition and axiom (postulate). Give a brief history of proofs and give some possible reasons for the Greek introduction of deduction (proofs) into mathematics.
• Most mathematical theorems are conditional statements: if p (hypothesis) then q (conclusion). There are three common techniques of proof; (i) Explain about direct proof (of if p then q). For an example, prove directly that "if n is an even integer, then n^2 is even." (ii) Explain about proof by contraposition. Prove by contraposition that "if m^2 is even, then m is even." (iii) Explain about proof by contradiction. Prove by contradiction that "if x is real and x^2 = 3, then x is irrational".
• Finally, write a detailed proof for the following problem. "Show that Playfair's Axiom is equivalent to Euclid's postulate 5, under the assumption that lines of arbitrary length may be drawn and therefore that Proposition I-16 is true."