Homework 7 –Average: 70Median: 75 – Homework 8 –Average: 74Median: 79 – –#36 was commonly missed The solution key presents a rather tedious, straightforward application of the counting techniques we discussed in class. There is a more clever way to solve this problem but it takes a bit of creative insight beyond what we discussed in class. I did not present this more clever solution in the key because I want you to see how it can be solved without this insight. Emmanuel’s last OH is this Thursday, April 24 th

Review for Final Exam (Part 1) Propositional Logic Know the rules and connectives Be able to translate statements into propositional logic Be able to produce logical equivalences Solve Logic Puzzles (Who’s Lying?, Knights/Knaves, etc.) Predicate Logic Know the rules, quantifies, and format for predicate logic Be able to translate statements into predicate logic Know how to form common statements (such as at least two, etc)

Methods of Proof Know how to apply the different proof techniques When can you use a counter-example, when do you need UG? Be able to prove given statements by applying the proper method Know how to spot invalid use of a proof technique Sets Know the definition of commonly used sets: N, Z, Z +, R, Q, etc. Be able to work with set operations: , , , , , P(S), etc. Have facility with set specifiers: =, , , , , |S|, etc. Know set terminology: empty, disjoint, etc. Understand the bit string representation of a subset of a given set Review things that gave you trouble on TEST1

Functions KNOW THE DEFINITION OF MAP AND FUNCTION! Be able to tell whether a function is 1-1 or onto and show it Understand function operations such as composition and inverse What is a 1-1 correspondence? How does it relate to counting? The Integers and Division Know the definitions: a | b, even, odd, rational, irrational, prime, composite, mod, relatively prime, pairwise relatively prime Review the theorems in this section regarding divisibility, mod, primes, fundamental theorem of arithmetic, division algorithm, etc. Be able to find the prime factorization, gcd, lcm, mod, etc. Prove results concerning divisibility, rational, irrational, prime, composite, mod, etc.

Proof Strategy The main goal is to be able to prove theorems. Strategies presented include working backwards from the desired result, using existing proofs, looking for counter-examples, making conjectures Sequences and Summations Review sequences, summations, summation notation, cardinality Know the definition of countability, uncountability, etc. How do sequences relate to countability? Be able to determine whether a set is countable or uncountable. Prove results concerning countability/uncountability. Understand the technique of diagonalization and how it is used to prove that a set is uncountable. Other methods of proving a set is uncountable?

Ex: Show that the set of irrational numbers is uncountable. We can not use the technique of diagonalization here because we don’t have a rigorous representation for irrational numbers. Proof: Assume, to the contrary, that the set of irrational numbers is countable. Use I to represent the set of irrational numbers. We know that R = Q  I. We also know that Q is countable. Hence R is countable as the union of two countable sets [we proved it]. But we know that R is uncountable [we proved this as well]. This is a contradiction: a set can’t be both countable and uncountable, so the set of irrational numbers must be uncountable, after all.  Take specific note here that diagonalization is not the only means by which to prove a set is countable. Also note that just because we may not be able to diagonalize a set doesn’t mean it isn’t uncountable.

Mathematical Induction Know the principle of mathematical induction When can we use mathematical induction Be able to prove results using mathematical induction Recursive Definitions and Structural Induction KNOW HOW A RECURSIVE SET DEFINITION WORKS! Be able to determine the elements of a recursively defined set Understand the technique of structural induction and how to use it to prove results about recursively defined sets and sequences. Review things that gave you trouble on TEST2 Pay specific attention to understanding countability/uncountability Also the workings of recursive set definition gave a lot of trouble.