Math 42, Discrete Mathematics

Transcription

1 c Fall 2018 last updated 12/10/2018 at 18:16:18 For use by students in this class only; all rights reserved. Note: some prose & some tables are taken directly from Kenneth R. Rosen, and Its Applications, 8th Ed., the ocial text adopted for this course.

2 As noted in class, this list is incomplete, and the absence of an item from the list does not preclude it's showing up on an examespecially if it's an item I've said is important. Probability Experiment, outcome, sample space, event. Equal likelihood probability. p(e) = E / S. Choosing at random" implies equal likelihood. Practical applications of equal likelihood probability employing, e.g., counting results from Chapter 6. Probabilities of complements: p(ē) = 1 p(e). Probabilities of unions: p(e F) = p(e) + p(f) p(e F). Basics of probability distributions p : S R. 0 p(s) 1 s S p(s) = 1 c R. P.

3 p(e) = s E p(s). p( ) = 0; p(s) = 1. If E F, then p(e) p(f). Mutually exclusive events E and F: E F =. p(e F) = 0. If E and F are mutually exclusive, then p(e F) = p(e) + p(f). Conditional probability: if p(f) 0, then p(e F) = p(e F). p(f) p(e F) = p(e F)p(F) c R. P.

4 Independence of E and F : if and only if p(e F) = p(e) if and only if p(e F) = p(e)p(f) if and only if p(f E) = p(f). Do not confuse mutually exclusive" with independent." mutually exclusive events are never independent. Don't use p(e F) = p(e)p(f) unless you know that E and F are independent events. Law of Total Probability: p(f) = p((f E) (F Ē)) = p(f E) + p(f Ē) = p(f E)p(E) + p(f Ē)p(Ē). (1) Note that the second equation in (1) holds because the union F = (F E) (F Ē) is a disjoint union. c R. P.

5 Tree diagrams and Venn diagrams to help remember/visualize the Law of Total Probability. Bayes' Theorem: p(e F) p(f E) p(f E)p(E) p(e F) = = = p(f) p(f) p(f) p(f E)p(E) = p(f E)p(E) + p(f Ē)p(Ē) (2) Note that we have used (1) in (2). Applications of the Law of Total Probability and of Bayes' Theorem to specic problems. c R. P.

6 Relations A (binary) relation R from a set A to a set B: R A B. Relation notation: if (a, b) R, we write arb and say that a is related to b by R. If (a, b) / R, we write a Rb. Examples of relations from A to B. A relation R on a set A: R A A, i.e., R is a relation from A to itself. Examples of relations on a set A. Properties of relations on A: Reexivity (R is reexive); examples. Symmetry (R is symmetric); examples. Transitivity (R is transitive); examples. Antisymmetry (R is antisymmetric); examples. c R. P.

7 Equivalence relations on A: relations that are reexive, symmetric, and transitive. Equivalence of elements of A: a 1 is equivalent to a 2 if and only if (a 1, a 2 ) R. Notation: a 1 a 2. Examples of equivalence relations. Proving/disproving that a relation is an equivalence relation. Congruence mod n: k l mod n if and only if n divides k l evenly. Equivalence classes: [a] = {b A b a} Representatives of equivalence classes: If b [a] then b is a representative of [a]. a b if and only if [a] = [b] if and only if [a] [b]. c R. P.

8 A partition of a set A is a collection of disjoint nonempty subsets of A whose union is all of A. In other words, P = {A i A i A i I} is a partition of A if and only if 1. A i i I; 2. A i A j = i j I; and 3. i I A i = A. Every equivalence relation on A yields a partition of A into its equivalence classes. Conversely, every partition gives rise to an equivalence relation. Examples of partitions. Representing a relation R on a set A = {a 1,..., a n } using a binary matrix M = [m ij ] as follows: { 1 if a i Ra j m ij = (3) 0 if a i Ra j. c R. P.

9 Examples of representing relations by matrices. Finding the original relation R on A = {a 1,..., a n }, given a binary matrix M. Detecting reexivity of the relation R, given the matrix M. Detecting symmetry of the relation R, given the matrix M. Detecting anti-symmetry of the relation R, given the matrix M. Computing M [2] = M M, the Boolean square of a binary matrix. Detecting transitivity of the relation R, using M [2] of the matrix M. Digraphs. Drawing the digraph of a relation R on A = {a 1,..., a n }. Finding the original relation R on A = {a 1,..., a n }, given a digraph. c R. P.