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Set Concepts

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Set Concepts

1. 1. Set Theory Professor Orr CPT120 ~ Quantitative Analysis I
2. 2. Why Study Set Theory? <ul><li>Understanding set theory helps people to … </li></ul><ul><li>see things in terms of systems </li></ul><ul><li>organize things into groups </li></ul><ul><li>begin to understand logic </li></ul>
3. 3. Key Mathematicians <ul><li>These mathematicians influenced the development of set theory and logic: </li></ul><ul><li>Georg Cantor </li></ul><ul><li>John Venn </li></ul><ul><li>George Boole </li></ul><ul><li>Augustus DeMorgan </li></ul>
4. 4. Georg Cantor 1845 -1918 <ul><li>developed set theory </li></ul><ul><li>set theory was not initially accepted because it was radically different </li></ul><ul><li>set theory today is widely accepted and is used in many areas of mathematics </li></ul>
5. 5. …Cantor <ul><li>the concept of infinity was expanded by Cantor’s set theory </li></ul><ul><li>Cantor proved there are “levels of infinity” </li></ul><ul><li>an infinitude of integers initially ending with  or </li></ul><ul><li>an infinitude of real numbers exist between 1 and 2; </li></ul><ul><li>there are more real numbers than there are integers… </li></ul>
6. 6. John Venn 1834-1923 <ul><li>studied and taught logic and probability theory </li></ul><ul><li>articulated Boole’s algebra of logic </li></ul><ul><li>devised a simple way to diagram set operations (Venn Diagrams) </li></ul>
7. 7. George Boole 1815-1864 <ul><li>self‑taught mathematician with an interest in logic </li></ul><ul><li>developed an algebra of logic (Boolean Algebra) </li></ul><ul><li>featured the operators </li></ul><ul><ul><li>and </li></ul></ul><ul><ul><li>or </li></ul></ul><ul><ul><li>not </li></ul></ul><ul><ul><li>nor (exclusive or) </li></ul></ul>
8. 8. Augustus De Morgan 1806-1871 <ul><li>developed two laws of negation </li></ul><ul><li>interested, like other mathematicians, in using mathematics to demonstrate logic </li></ul><ul><li>furthered Boole’s work of incorporating logic and mathematics </li></ul><ul><li>formally stated the laws of set theory </li></ul>
9. 9. Basic Set Theory Definitions <ul><li>A set is a collection of elements </li></ul><ul><li>An element is an object contained in a set </li></ul><ul><li>If every element of Set A is also contained in Set B , then Set A is a subset of Set B </li></ul><ul><ul><li>A is a proper subset of B if B has more elements than A does </li></ul></ul><ul><li>The universal set contains all of the elements relevant to a given discussion </li></ul>
10. 10. Simple Set Example <ul><li>the universal set is a deck of ordinary playing cards </li></ul><ul><li>each card is an element in the universal set </li></ul><ul><li>some subsets are: </li></ul><ul><ul><li>face cards </li></ul></ul><ul><ul><li>numbered cards </li></ul></ul><ul><ul><li>suits </li></ul></ul><ul><ul><li>poker hands </li></ul></ul>
11. 11. Set Theory Notation <ul><li>Symbol Meaning </li></ul><ul><li>Upper case designates set name </li></ul><ul><li>Lower case designates set elements </li></ul><ul><li>{ } enclose elements in set </li></ul><ul><li> or is (or is not) an element of </li></ul><ul><li> is a subset of (includes equal sets) </li></ul><ul><li> is a proper subset of </li></ul><ul><li> is not a subset of </li></ul><ul><li> is a superset of </li></ul><ul><li>| or : such that (if a condition is true) </li></ul><ul><li>| | the cardinality of a set </li></ul>
12. 12. Set Notation: Defining Sets <ul><li>a set is a collection of objects </li></ul><ul><li>sets can be defined two ways: </li></ul><ul><ul><li>by listing each element </li></ul></ul><ul><ul><li>by defining the rules for membership </li></ul></ul><ul><li>Examples: </li></ul><ul><ul><li>A = {2,4,6,8,10} </li></ul></ul><ul><ul><li>A = { x | x is a positive even integer <12} </li></ul></ul>
13. 13. Set Notation Elements <ul><li>an element is a member of a set </li></ul><ul><li>notation:  means “is an element of”  means “is not an element of” </li></ul><ul><li>Examples: </li></ul><ul><ul><li>A = {1, 2, 3, 4} </li></ul></ul><ul><li> 1  A 6  A </li></ul><ul><li> 2  A z  A </li></ul><ul><ul><li>B = {x | x is an even number  10} </li></ul></ul><ul><li> 2  B 9  B </li></ul><ul><li> 4  B z  B </li></ul>
14. 14. Subsets <ul><li>a subset exists when a set’s members are also contained in another set </li></ul><ul><li>notation:  means “is a subset of”  means “is a proper subset of”  means “is not a subset of” </li></ul>
15. 15. Subset Relationships <ul><li>A = {x | x is a positive integer  8} </li></ul><ul><li>set A contains: 1, 2, 3, 4, 5, 6, 7, 8 </li></ul><ul><li>B = {x | x is a positive even integer  10} </li></ul><ul><li>set B contains: 2, 4, 6, 8 </li></ul><ul><li>C = {2, 4, 6, 8, 10} </li></ul><ul><li>set C contains: 2, 4, 6, 8, 10 </li></ul><ul><li>Subset Relationships </li></ul><ul><li>A  A A  B A  C </li></ul><ul><li>B  A B  B B  C </li></ul><ul><li>C  A C  B C  C </li></ul>
16. 16. Set Equality <ul><li>Two sets are equal if and only if they contain precisely the same elements. </li></ul><ul><li>The order in which the elements are listed is unimportant. </li></ul><ul><li>Elements may be repeated in set definitions without increasing the size of the sets. </li></ul><ul><li>Examples: </li></ul><ul><li>A = {1, 2, 3, 4} B = {1, 4, 2, 3} </li></ul><ul><li>A  B and B  A; therefore, A = B and B = A </li></ul><ul><li>A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4} </li></ul><ul><li>A  B and B  A; therefore, A = B and B = A </li></ul>
17. 17. Cardinality of Sets <ul><li>Cardinality refers to the number of elements in a set </li></ul><ul><li>A finite set has a countable number of elements </li></ul><ul><li>An infinite set has at least as many elements as the set of natural numbers </li></ul><ul><li>notation: |A| represents the cardinality of Set A </li></ul>
18. 18. Finite Set Cardinality <ul><li>Set Definition Cardinality </li></ul><ul><li>A = {x | x is a lower case letter} |A| = 26 </li></ul><ul><li>B = {2, 3, 4, 5, 6, 7} |B| = 6 </li></ul><ul><li>C = {x | x is an even number  10} |C|= 4 </li></ul><ul><li>D = {x | x is an even number  10} |D| = 5 </li></ul>
19. 19. Infinite Set Cardinality <ul><li>Set Definition Cardinality </li></ul><ul><li>A = {1, 2, 3, …} |A| = </li></ul><ul><li>B = {x | x is a point on a line} |B| = </li></ul><ul><li>C = {x| x is a point in a plane} |C| = </li></ul>
20. 20. Universal Sets <ul><li>The universal set is the set of all things pertinent to to a given discussion and is designated by the symbol U </li></ul><ul><li>Example : </li></ul><ul><li>U = {all students at IUPUI} </li></ul><ul><li>Some Subsets : </li></ul><ul><li>A = {all Computer Technology students} </li></ul><ul><li>B = {freshmen students} </li></ul><ul><li>C = {sophomore students} </li></ul>
21. 21. The Empty Set <ul><li>Any set that contains no elements is called the empty set </li></ul><ul><li>the empty set is a subset of every set including itself </li></ul><ul><li>notation: { } or  </li></ul><ul><li>Examples ~ both A and B are empty </li></ul><ul><li>A = {x | x is a Chevrolet Mustang} </li></ul><ul><li>B = {x | x is a positive number  0} </li></ul>
22. 22. The Power Set ( P ) <ul><li>The power set is the set of all subsets that can be created from a given set </li></ul><ul><li>The cardinality of the power set is 2 to the power of the given set’s cardinality </li></ul><ul><li>notation: P ( set name) </li></ul><ul><li>Example: </li></ul><ul><li>A = {a, b, c} where |A| = 3 </li></ul><ul><li>P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A,  } and | P (A)| = 8 </li></ul><ul><li>In general, if |A| = n , then | P (A) | = 2 n </li></ul>
23. 23. Special Sets <ul><li>Z represents the set of integers </li></ul><ul><ul><li>Z + is the set of positive integers and </li></ul></ul><ul><ul><li>Z - is the set of negative integers </li></ul></ul><ul><li>N represents the set of natural numbers </li></ul><ul><li>ℝ represents the set of real numbers </li></ul><ul><li>Q represents the set of rational numbers </li></ul>
24. 24. Venn Diagrams <ul><li>Venn diagrams show relationships between sets and their elements </li></ul>Universal Set Sets A & B
25. 25. Venn Diagram Example 1 <ul><li>Set Definition Elements </li></ul><ul><li>A = {x | x  Z + and x  8} 1 2 3 4 5 6 7 8 </li></ul><ul><li>B = {x | x  Z + ; x is even and  10} 2 4 6 8 10 </li></ul><ul><li>A  B </li></ul><ul><li>B  A </li></ul>
26. 26. Venn Diagram Example 2 <ul><li>Set Definition Elements </li></ul><ul><li>A = {x | x  Z + and x  9} 1 2 3 4 5 6 7 8 9 </li></ul><ul><li>B = {x | x  Z + ; x is even and  8} 2 4 6 8 </li></ul><ul><li>A  B </li></ul><ul><li>B  A </li></ul><ul><li>A  B </li></ul>
27. 27. Venn Diagram Example 3 <ul><li>Set Definition Elements </li></ul><ul><li>A = {x | x  Z + ; x is even and  10} 2 4 6 8 10 </li></ul><ul><li>B = x  Z + ; x is odd and x  10 } 1 3 5 7 9 </li></ul><ul><li>A  B </li></ul><ul><li>B  A </li></ul>
28. 28. Venn Diagram Example 4 <ul><li>Set Definition </li></ul><ul><li> U = {1, 2, 3, 4, 5, 6, 7, 8} </li></ul><ul><li>A = {1, 2, 6, 7} </li></ul><ul><li>B = {2, 3, 4, 7} </li></ul><ul><li>C = {4, 5, 6, 7} </li></ul>A = {1, 2, 6, 7}
29. 29. Venn Diagram Example 5 <ul><li>Set Definition </li></ul><ul><li> U = {1, 2, 3, 4, 5, 6, 7, 8} </li></ul><ul><li>A = {1, 2, 6, 7} </li></ul><ul><li>B = {2, 3, 4, 7} </li></ul><ul><li>C = {4, 5, 6, 7} </li></ul>B = {2, 3, 4, 7}
30. 30. Venn Diagram Example 6 <ul><li>Set Definition </li></ul><ul><li> U = {1, 2, 3, 4, 5, 6, 7, 8} </li></ul><ul><li>A = {1, 2, 6, 7} </li></ul><ul><li>B = {2, 3, 4, 7} </li></ul><ul><li>C = {4, 5, 6, 7} </li></ul>C = {4, 5, 6, 7}