Homework 9 –Average: 81Median: 88 – –Section 7.3, #9 Note that the answers given in the back of the text were incorrect for parts (d) and (e). I gave and graded this problem because I was interested to see how many of you were just copying the solutions from the text for odd numbered problems instead of working the problems yourselves and using the answers given to verify yours. A large portion of the class got (d) and (e) incorrect. Homework 10 (Due Today) –You will receive a participation grade (4pts per problem) –

Review for Final Exam (Part 2) [This also serves as review for Test3] The Basics of Counting Know the product rule, sum rule, and principle of inclusion- exclusion and how to apply them and when they can be applied. Ex: (Section 4.1, #42) How many bit strings of length 10 contain either five consecutive 0s or five consecutive 1s? First count the number of bit strings that contain 5 consecutive 0s: 00000xxxxx = xxxx = 2 4 x100000xxx = 2 4 xx100000xx = 2 4 xxx100000x = 2 4 xxxx = 2 4 So the number of bit strings of length 10 that contain five consecutive 0s is *2 4 = 112. Similarly the number of bit strings of length 10 that contain five consecutive 1s is 112. But and are d.c.ed. So – 2 = 222.

The Pigeonhole Principle Know how to apply the pigeonhole principle and the GPP. Permutations and Combinations Know the definition of permutation, combination, r-permutation, and r-combination and how they are used in counting. Ex: (Section 4.3, #36) How many bit strings contain exactly five 0s and 14 1s if every 0 must be immediately followed by two 1s? Here we’ll use a more clever technique to solve this problem. Consider just as before that we are looking for strings of the form: x011x011x011x011x011x with 4 1’s left to place among the x’s. Instead of considering cases, let’s consider that we have 5 copies of 011 to place and 4 copies of 1 to place. So we can think of this as having 9 blanks where we have to put 011 into 5 of them and 1 into 4. C(9, 5) or C(9, 4) = 9!/[5!4!] = [9*8*7*6]/[4*3*2*1] = 3*7*6 = 126.

Relations and Their Properties Know the definition of relation (especially relation on a set) Know the definitions of the six properties we studied and how to determine whether a relation has each of the properties. Review theorems regarding relations and their properties Representing Relations Be comfortable with the 3 different ways we have to represent relations. They are each very useful for determining whether a relation has certain properties, computing closures, etc. Be able to translate between different representations of a relation. Closures of Relations Know the definition of closure and how to find closures KNOW HOW TO FIND THE TRANSITIVE CLOSURE (paths)

Equivalence Relation Know the definition of Equivalence Relation, how to prove it Know the definition of Equivalence Classes of an Equivalence Relation, how to find them, and theorems about them Know the definition of a partition and relationship to equiv. rels. Theorem 2 of 7.5 details the relationship between equivalence classes and partitions. There is a 1-1 correspondence. This means if you are given a partition over a set A then there is a corresponding equivalence relation over A such that the equivalence classes of this relation are the sets of the partition. Conversely, given an equivalence relation over a set A, if you put its equivalence classes into a set, you get a partition of A. If I give you an equivalence relation over a set A, you need to be able to find the partition it gives rise to. If I give you a partition over A, you need to find the eq. rel.

Partial Orderings Know all definitions from this section: partial ordering, poset, comparable, incomparable, total order, chain, well-ordering, etc. Be able to construct a Hasse diagram for a given poset. Know all definitions related to Hasse diagrams: maximal/minimal element, greatest/least elements, upper/lower bound, lub/glb, etc. Determine whether a poset is a lattice. For TEST3, review things that gave you trouble in the HW’s (8, 9, 10) For the Final, review things that give you trouble on TEST3 Chapter 4 is very problem solving oriented. Counting is a form of problem solving and you need to develop these techniques through practice. Always ask: “What am I supposed to count?”, “Am I counting everything I need to?”, “Am I double-counting things?” Chapter 7 is extremely definition oriented. You must know them!