Discrete Structures for Computer Science
Ruoming Jin MW 2:15 3:00pm Spring 2010 rm MSB115
Course Material
Textbook: Discrete Mathematics and Its Applications Kenneth H. Rosen, McGraw Hill
Course Requirements
Homework, 20% Quiz, 20% Three Intermediate Ex
R e la tio ns
Binary Relations
a re la tio nb e twe e ne le m e nts o ftwo s e ts is a s ub s e t o fth e irC a rte s ia np ro d uc t(o fo rd e re d p a irs ) . No te th e d ie re nc e b e twe e na re la tio na nd a func tio n:ina re la tio n,e a c h a A
Induction and Recursion
Odd Powers Are Odd
Fact: If m is odd and n is odd, then nm is odd. Proposition: for an odd number m, mk is odd for all non-negative integer k.
Let P(i) be the proposition that mi is odd.
Proof by induction
P(1) is true by definiti
Counting
Counting in Algorithms
How many comparisons are needed to sort n numbers?
How many steps to compute the GCD of two numbers ?
How many steps to factor an integer?
Counting in Games
How many different configurations for a Rubiks cube?
How many
Functions
f( ) =
A
Lecture 12
B
Functions
f :A B
function, f, from set A to set B associates an element
f (a) B
, with an element
a A.
The domain of f is A. The codomain of f is B.
For every input there is exactly one output.
Functions
Functions f(S) = |S
Sets
1
Sets
Informally:
A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, integers,
complex numbers, C All students in our class
Defining Sets
Sets can be defined directl
Logic and Proof
Argument
An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid if: whenever all the assumptions are true, then the
First Order Logic
Propositional Logic
A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions?
Toronto is the capital of Canada. Read this carefully.