1. Discrete Structures Review Presented by: Aaron Morales Tony Thi

2. Math Logic In Math Logic (Propositional Calculus) we use connectives and variables to create statements. Connectives are: not (negation)~~X and (conjuction)^X ^ Y or (disjunction)VX V Y Implies (conditional)  X  Y IFF (biconditional)  X  Y

3. Truth Tables and Definitions XY~XX ^ YX V YX --> YX Y TTFTTTT TFFFTFF FTTFTTF FFTFFTT Two statements are logically EQUIVALENT if they have the same truth table A Statement is a TAUTOLOGY if its truth table consists entirely of ‘Ts’ A well formed proposition (WFP) is a valid statement e.g. X ^ Y is WFP, X ^  Y is not

4. Boolean Functions and Logic Gates Variables are streams of 0s and 1s True  1 and False  0 Connective notation changes AND ABORA + B NOTĀ XOR A (+) B Logic Gates are icons for boolean operations e.g. NAND, NOR DeMorgan’s Law: ______ ĀĒ = A + E AE = Ā + Ē

5. Elementary Combinatorics Fundamental Principle of counting How many 3 digit odd numbers are there that are less than 700? Unordered samples w/o repetition, is given by the binomial coefficient n over k (n / k) calculated by n! / k!(n-k!). e.g. How many 8-bit binary numbers contain exactly 3 ones? (8/3) = 56 notice 2^8 = 256 Unordered samples w/repetition, is given by the binomial coefficient n+k-1 over k, (n+k-1 / k) e.g. suppose we have four different donuts, in how many different ways can we select a dozen unordered w/repetition? n=4, k=12, (15 / 12) = (15 / 3) = 455

6. Graph Theory A graph is a set of points (vertices) together with a set of lines (edges) joining some of these points. The degree of a vertex is the number of edges connected to it. A cycle is the path ABC Graph 2 is a tree, connected and no loops

7. MinCost Spanning Tree Given a connected graph G, a spanning tree, T, T subset G A MCST for a graph is a spanning tree for which the sum of the cost is a minimum By Prim’s Algorithm -Start at random - Choose the smallest cost -Connect an edge that is not in the tree but is the smallest cost e.g. 6+2+4+5+5 = 22 Dijkstra’s Algorithm?

8. Binary Tree A binary tree is a directed graph (downwards) in which - the in-degree of exactly one vertex (root) is zero - the in-degree of all other vertices is 1 - the out-degree of each vertex is either 0, 1, or 2 What is a sub tree of a binary tree? What is Pre-Order Search? RoLR What is Post-Order Search? LRRo

9. Textbook Introduction to Discrete Mathematics by McEliece, Ash, and Ash (Random House Publishers)