Review

1
Review

• CS311
• Western Washington University

2
Terminology

• Converse
• the converse of If A then B is If B then A
• Contrapositive
• the contrapositive of If A then B is If not B,
  then not A

3
Proof Techniques

• Proof by Example
• There is a prime number between 80 and 88
• Proof by Exhaustive Checking
• The sum of any of the odd numbers 1, 3, and 5 is
  an even number
• Proof Using Variables
• The sum of any two odd integers is even
• Direct Proof
• If A then B
• assume A is true to prove B
• Indirect Proof
• If A then B
• prove the contrapositive If not B, then not A
• proof by contradiction
• assume the entire statement is false

4

• If and Only If Proofs
• requires proving the conditional AND its converse
• Proof by Induction
• Needed to prove properties of functions and
  procedures
• To prove P(n), first prove P(0), then assume P(k)
  is true to prove that P(k1) is true
• P(N) gt 1 2 3 n n(n1)/2
• P(0) gt 0 0(01)/2 0
• Assume P(k) gt 12kk(k1)/2
• Prove P(k1)
• 12kk1(12k)(k1) by the associative
  property of addition
• k(k1)/2 (k1) by assumption
• (k1)((k1) 1)/2 by factoring

5
Sets

• A set is a collection of things called its
  elements, members, or objects
• Set notation and terms
• membership
• listing
• empty set
• equality
• inequality
• subset
• S x x has property P
• power set
• Natural numbers
• Integers
• Rational numbers
• Venn diagram

6
Set Operations

• Union
• Intersection
• Disjoint
• Difference
• Symmetric difference (Exclusive OR)
• Cardinality

7
Sets, Bags, Tuples

• Sets
• h, u, g, h h, u, g u, g, h
• duplicates are ignored
• order is unimportant
• Bags
• h, u, g, h h, h, u, g , but does not equal
  h, u, g
• duplicates are not ignored
• order is unimportant
• Tuple
• lth, u, g, hgt does not equal lth, h, u, ggt
• order matters
• Products of Sets
• A x B lta,bgt a is an element of A, and b is
  an element of B

8
Function Terminology

• A -gt B denotes the set of all functions from A to
  B
• f is a function from A to B
• fA-gtB
• A is the domain of f
• B is the codomain of f
• the range of f is the set of elements in B that
  are associated with some element in A f(a) a
  ? A
• Partial function
• a partial function from A to B may not be defined
  for some elements of A
• I.e. division for Rationals
• Composition
• fA-gtB and gB-gtC
• g?fA-gtC

9
Languages

• A language is a set of strings
• An alphabet is the set of elements used to
  compose the strings of a language
• A grammar is a set of rules defining the
  structure of the strings in a language
• 4 parts
• N - an alphabet of grammar symbols called non
  terminals
• T - an alphabet of symbols called terminals
• S - a non terminal starting point
• P - production rules of the form A-gtB where A and
  B are strings over the alphabet N union T