Lecture 04 Set Theory Profs. Koike and Yukita

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Presentation transcript:

Lecture 04 Set Theory Profs. Koike and Yukita Discrete Systems I Lecture 04 Set Theory Profs. Koike and Yukita

1. Goals (You will be familiar with ) Notation and Terminology of Set Theory. Mathematical Induction. Disc Sys 04

2. Sets and Elements notation Disc Sys 04

How to read “p is an element of A” “p is not an element of A” “p belongs to A” “p is not an element of A” “p does not belong to A” A is the set whose elements are a, i, u, e, and o. B is the set of all posive even integers. Disc Sys 04

Convention Use capital letters A, B, … to denote sets. Use lower case letters a, b, … to denote elements of sets. Disc Sys 04

The following sets are all equal. Disc Sys 04

Special symbols for some sets Disc Sys 04

3. Universal set and Empty set The universal set The empty set Disc Sys 04

4. Subsets notation, how to read A is a subset of B. B contains A. A is contained in B. Disc Sys 04

Examples of inclusion Disc Sys 04

Proper subset Disc Sys 04

5. Venn diagrams U B A Disc Sys 04

Disjoint sets U A B Disc Sys 04

Intersecting sets U B A Disc Sys 04

6. Set operations Union and Intersection Disc Sys 04

Complements The complement of A is the set of elements which belong to U but which do not belong to A. U A Disc Sys 04

Difference, Relative complement A minus B. A B Disc Sys 04

7. Algebra of Sets and Duality Trivial rules: Disc Sys 04

Distributive Laws A A B B C C Disc Sys 04

De Morgan's Laws U U B B A A Disc Sys 04

Duality To get the dual of an equation of set algebra, replace each occurrence of by Disc Sys 04

8. Counting Principle for finite sets Disc Sys 04

9. Power sets Given a set S, we call the set of all subsets of S the power set of S and denote it by Disc Sys 04

The power set of Disc Sys 04

Generalized set operations Disc Sys 04

Examples Disc Sys 04

10. Mathematical Induction Let P be the proposition defined on the positive integers N, i.e., P(n) is either true or false for each n in N. (i) P(1) is true. (ii) P(n+1) is true whenever P(n) is true. P(n) is true for every positive integer n. Disc Sys 04

Problem 1 Consider the Venn diagram of two arbitrary sets A and B. Shade the sets: U A B 1.10 Disc Sys 04

Problem 2 Illustrate the distributive law with Venn diagrams. A B C 1.11 Disc Sys 04

Problem 3 Illustrate the distributive law with Venn diagrams. A B C 1.11 Disc Sys 04

Problem 4 1.17 Disc Sys 04

Hints for a rigorous approach to Problem 4 Disc Sys 04

Problem 5 1.20 Disc Sys 04

Problem 6 1.24 Disc Sys 04