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

2. 1. Goals (You will be familiar with )
Notation and Terminology of Set Theory.
Mathematical Induction.
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3. 2. Sets and Elements notation
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4. 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.
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5. Convention Use capital letters A, B, … to denote sets.
Use lower case letters a, b, … to denote elements of sets.
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6. The following sets are all equal.
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7. Special symbols for some sets
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8. 3. Universal set and Empty set
The universal set
The empty set
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9. 4. Subsets notation, how to read
A is a subset of B.
B contains A.
A is contained in B.
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10. Examples of inclusion
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11. Proper subset
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12. 5. Venn diagrams U B A
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13. Disjoint sets U A B
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14. Intersecting sets U B A
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15. 6. Set operations Union and Intersection
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16. Complements
The complement of A is the set of elements which belong to U but which do not belong to A. U A
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17. Difference, Relative complement
A minus B. A B
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18. 7. Algebra of Sets and Duality
Trivial rules:
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19. Distributive Laws A A B B C C
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20. De Morgan's Laws U U B B A A
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21. Duality To get the dual of an equation of set algebra,
replace each occurrence of by
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22. 8. Counting Principle for finite sets
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23. 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
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24. The power set of
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25. Generalized set operations
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26. Examples
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27. 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.
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28. Problem 1
Consider the Venn diagram of two arbitrary sets A and B. Shade the sets: U A B
1.10
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29. Problem 2 Illustrate the distributive law with Venn diagrams. A B C
1.11
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30. Problem 3 Illustrate the distributive law with Venn diagrams. A B C
1.11
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31. Problem 4
1.17
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32. Hints for a rigorous approach to Problem 4
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33. Problem 5
1.20
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34. Problem 6
1.24
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