Discrete Structures Review

Presented by:Aaron MoralesTony Thi

Math Logic

In Math Logic (Propositional Calculus) we use connectives and variables to create statements.

Connectives are:not (negation) ~ ~Xand (conjuction) ^ X ^ Yor (disjunction) V X V YImplies (conditional) X YIFF (biconditional) X Y

Truth Tables and Definitions

X Y ~X X ^ Y X V Y X --> Y X <--> Y

T T F T T T T

T F F F T F F

F T T F T T F

F F T F F T T

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

Boolean Functions and Logic Gates

Variables are streams of 0s and 1s True 1 and False 0 Connective notation changes

AND AB OR A + BNOT Ā XOR A (+) B

Logic Gates are icons for boolean operations e.g. NAND, NOR

DeMorgan’s Law: ____ __

ĀĒ = A + E AE = Ā + Ē

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

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

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 coste.g. 6+2+4+5+5 = 22

Dijkstra’s Algorithm?

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

Textbook

Introduction to Discrete Mathematicsby McEliece, Ash, and Ash(Random House Publishers)