1. Information Safe
Background
Hoang V.Nguyen
Mail: startnewday85@gmail.com
Department of Computer Science
Faculty of Information Technology – Hanoi University of Agriculture

2. Background
• Set theory and functions
• Probability and Information theory
• Complexity theory
• Abstract Algebra
• Number theory

3. Set theory and functions
• Set theory
A B
• Functions
- Relations: 1-1, 1-n, n-1, n-n
- functions:
• domain vs codomain
• preimage vs image
• injective, onto, bijective
• inverse
• permutation A B

4. Probability theory
• a language for “randomness”
• How:
- experiment
- simple events, event
- sample space: discrete vs continous
- probability distribution
- conditional probability and Bayes’ theorem
- random variables
- expected value or mean
- cumulative distribution function, probability function,
density function
- some distributions: normal, binomial, … .

5. Information theory
• What’s information, how to measure?
• 1940s, Claude Shannon: Entropy
n
H ( A)   pi log pi
i 1

6. Complexity theory
• What are the mathematical models of
computation?
Automata theory
• What problems can(not) computers solve?
Computability theory
• What makes some problems computationally
hard and other easy?
Complexity theory

7. Complexity theory
• Rate of growth/Order of growth
• Classify problems
- P, NP, NPC, NP_hard
- PSPACE
- FP

8. Abstract Algebra
• Group
- is a set with a binary operation such that: associate, identity
element and inverse element
- is abelian group: cummutative
- subgroup, generator element, cyclic group
• Ring
-is a set(R) with two binary operations(+, x) such that:
-(R,+) is abelian group with identify denoted by 0
-x operation is associate with identify is 1 ≠ 0
-The operation x is distributive over + operation
- commutative ring: x operation is commutative
• Field
- is a commutative ring which all non-zero elements have multiplicative inverses

9. Number theory
• What’s number?
• Characters of numbers
• Integer numbers
- divisibility
- prime numbers
- modulo