Topics for Major Examination III

1. The components of a mathematical system.
a. What are primitive terms?
b. What are axioms?
c. Give an example of an axiom.
d. Give the primitive terms in your example of an axiom.
e. What is the difference between a conjecture and a theorem?
2. Give truth tables

3. Implementations

a. Give the hypothesis
b. Give the conclusion.
4. Give a Venn diagram representing A and B.

5. Direct proofs

a. Describe the concept of a direct proof.
b. Prove a statement using direct proof.
6.  What is a syllogism?
a.  Give an example
b.  Define
c.  Give the conclusion given the premises of a syllogism.
8.  Contrapositive
a.  Define contrapositive
b.  Prove, using a truth table, that a statement and its contrapositive are equivalent
c.  Give a proof of a statement by proving its contrapositive.
d.  Give the contrapositive of a statement.
9.  Prime numbers
a.  Define prime numbers
b.  Determine if a number is prime.
c.  Give the prime factorization of a number
10.  Proof by contradiction
a.  What is the principle of proof by contradiction?
b.  Why does proof by contradiction work?
c.  Prove that there is no largest prime number
d.  Prove that the square root of a number is irrational
e.  Prove that the cube root of a number is irrational
f.  Show why the proof technique of part d does not work if the number is a perfect square.
11.  Qualified Statements
a.  Be able to express a qualified statement in symbols.
b.  Be able to give the negation of qualified statements
12.  Mathematical Induction
a.  Understand the concept of summation.
b.  Be able to define the principle of mathematical induction
c.  Give proofs of summation formulas using mathematical induction