1. 1 ENM 503 Block 1 Algebraic Systems Lesson 2 – The Algebra of Sets The Essence of Sets What are they?

2. 2 Set Theory Theory: A formal mathematical system consisting of a set of axioms and the rules of logic for deriving theorems from those axioms. Set theory – a branch of abstract mathematics set – a concept so basic that it is an undefined term consider a set a well-defined collection of objects that are called the elements of the set

3. 3 What is a set? A set is a collection of things Each entry in a set is known as an element. Sets are written using brackets { } with their elements listed in between For example the English alphabet could be written as: {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} even numbers could be {0,2,4,6,8,10,...} (Note: the dots at the end indicating that the set goes on infinitely)

4. 4 Set Equality Definition (Equality of sets): Two sets are equal if and only if they have the same elements. More formally, for any sets A and B, A = B if and only if  x  x  A, then x  B. Thus for example {1, 2, 3} = {3, 2, 1}, that is the order of elements does not matter, and {1, 2, 3} = {3, 2, 1, 1}, that is duplications do not make any difference for sets. (note:  reads “for all” and  reads “such that”)

5. 5 Subsets Definition (Subset): A set A is a subset of a set B if and only if everything in A is also in B. More formally, for any sets A and B, A is a subset of B, and denoted by A  B, if and only if  x  A, then x  B If A  B, and A  B, then A is said to be a proper subset of B and it is denoted by A  B. For example {1, 2}  {3, 2, 1}. Also {1, 2}  {3, 2, 1}. If A  B and B  C, then A  C

6. 6 More to do with subsets If even one element of one set is not contained within the other then they are not subsets. If A were defined as {1,2,3,4,5} and B as {3,4,5,6} then B would not be a subset of A since 6  B but 6  A. The symbol for “not a subset” is . We would write B  A.

7. 7 Some Examples b  {a, b, c, d} e  {a, b, c, d} {1, 2, 4, 5} = {2, 1, 5, 4} {1, 2, 3}  {1, 2, 3} {1, 2, 3}  {1, 2, 3, 4} {1, 2, 3}  {1, 2, 3, 4} {1, 2, 3,..., 1000} is a finite set. {1, 2, 3,...} is an infinite set.

8. 8 Two sets of note: The set containing no elements is the “empty” or “null” set and is denoted by { } or  All sets under consideration are regarded as subsets of a fixed set known as the Universal set and denoted by U

9. 9 Venn Diagrams Venn Diagrams were first developed by John Venn in the 1880s. They are useful for illustrating the relationships among elements in a set. For example if we want to represent the set of all counting numbers, and illustrate how even numbers and multiples of 3 are related, we could draw the following picture: The Universe

10. 10 Boolean Algebra A Boolean algebra is an algebra in which binary operations are chosen to model mathematical or logical operations in Set Theory. Specifically, for any sets A and B, it deals with the set operations of intersections and unions or the logic operations of “AND” and “OR” Also includes negation or the complement - the logic operation of “NOT.”

11. 11 Union A union of two or more sets is another set that contains everything contained in the previous sets. Union is designated by the symbol . If A and B are sets then A  B represents the union of A and B The union of A and B is the set of all elements that are either in A or B (or both), therefore A  B = {x | x  A or x  B}. “OR” logic “such that”

12. 12 Examples of the union of two sets Example 1: A={1,2,3,4,5}; B={5,7,9,11,13} A  B = {1,2,3,4,5,7,9,11,13} Example 2: A={all the books written by Charles Dickens} B={all the books written by Mark Twain} A  B = {all books written by either Charles Dickens or Mark Twain}

13. 13 Intersection The intersection of two (or more) sets is those elements that they have in common. Intersection is designated by the symbol . So if A and B are sets then the intersection is denoted by A  B. The intersection of A and B is the set of all elements that are common to A and B, therefore A  B = {x | x  A and x  B} “And” logic

14. 14 Examples of the intersection of two sets Example 1 A={1,3,5,7,9}; B={2,3,4,5,6} The elements they have in common are 3 and 5 A  B = {3,5} Example 2 A={The English alphabet} B={vowels} So A  B = {vowels} Example 3 A={1,2,3,4,5} B={6,7,8,9,10} In this case A and B have nothing in common. A  B =  is called the “empty or null set.”  = { }

15. 15 Negation or Complement Given the set A, then the set A' is the complement of A consisting of all elements not in A; i.e. A’ = {x| x  A} A A’ Let U = universal set (the set of all objects under discussion) Then A  A’ = U and A  A’ =  (they are mutually exclusive) Sometimes the complement of a set A is written as A c U

16. 16 Mutual Exclusive Sets Two sets are mutually exclusive (also called disjoint) if they do not have any elements in common; they need not together comprise the universal set. The following Venn diagram represents mutually exclusive (disjoint) sets. A  B = 

17. 17 A Partition A partition of a set S is a subdivision of S into subsets which are disjoint and whose union is S. That is, each x  S belongs to one and only one of the subsets If A 1, A 2, …, A n form a partition, then A 1  A 2  …  A n = S, and for any A i and A j i  j, A i  A j =  Often the set S = U

18. 18 The four disjoint regions of two intersecting sets forming a partition The four regions in which two circles divide the universal set can be identified as intersections of the two subsets and their complements as labeled in the following Venn diagram. U = (A’  B’)  (A  B’)  (A  B)  (A’  B)

19. 19 Three intersecting sets

20. 20 SET THEOREMS If A  B then A  B = A and A  B = B Quick student exercise: Create an example to illustrate each theorem. Did you know: (A  B)  A  (A  B) ?

21. 21 Examples Let S be the set of all integers, and let A = {2, 4, 6, 8} B = {5, 6, 7, 8} C = {positive even integers} D = {1, 2, 3}. Then A  B = {2, 4, 5, 6, 7, 8} A  B = {6, 8} A  C = A C ' = {0, 1, - 1, - 2, 3, - 3, - 4, 5, - 5,...} A  (B  C) = A

22. 22 Closure If A and B are any two sets then A  B is a set A  B is a set A’ is a set I get it. The algebra of sets is closed under the operations of union, intersection, and complements.

23. 23 The Laws of the Algebra of Sets De Morgan's Laws: (A  B)' = A'  B‘ (A  B)' = A'  B‘ Idempotent Laws: (A  A) = A (A  A) = A Associative Laws: (A  B)  C = A  (B  C) (A  B)  C = A  (B  C) Commutative Laws: (A  B)= (B  A) (A  B) = (B  A) Distributive Laws: A  (B  C) = (A  B)  C) A  (B  C) = (A  B)  C) Identity Laws (A   ) = A (A  U) = U (A   ) =  (A  U) = A Complement Laws (A  A’) = U (A’)’ = A (A  A’) =  U’ = ,  ’ = U These are very good laws. Quick student exercise: Prove using Venn diagrams

24. Set Algebra – some examples 24 

25. 25 Cartesian Product The Cartesian product of two sets, A and B, is the set of all ordered pairs (a, b) with a  A and b  B. A x B = { (a, b) | a  A and b  B }. A x B is the set of all ordered pairs whose first component is in A and whose second component is in B.

26. 26 Cartesian Product - examples 1. If A = {a, b} and B = {1, 2, 3}, then A x B = { (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) }. 2. Let S = {H, T}. (H stands for Heads, T stands for Tails) S x S = { (H,H), (H,T), (T,H), T,T) }. If S is the set of outcomes of tossing a coin once, then S is the set of outcomes of tossing a coin twice.

27. 27 Yet another example S = {1, 2, 3, 4, 5, 6} The set of outcomes of rolling a die S x S = {(1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6), (2, 1),(2, 2),(2, 3),(2, 4),(2, 5),(2, 6),(3, 1),(3, 2),(3, 3),(3, 4),(3, 5),(3, 6),(4, 1),(4, 2),(4, 3),(4, 4),(4, 5),(4, 6),(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6),(6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6)}

28. 28 A Cartesian Product Theorem If A, B, and C are any sets, then (A  B) x C = (A x C)  (B x C) Quick student exercise: Demonstrate the truth of this theorem by creating an example. That is, show both sets have the same ordered pairs. I get it. It is a type of distributive law for the Cartesian product. Another quick student exercise: Demonstrate that A x (B  C) = (A x B)  (A x C) is true or not true.

29. 29 Is there any more of this stuff? I can see where we can form new sets from unions, intersections, complements, and cross products of other sets. Are there any other ways of generating new sets? An engineering management student on the job.

30. 30 Power Sets Let U be the universal set. Then the set whose elements are all the subsets in U is called the power set of U and is denoted by P U. There are 2 n elements in the set P U where n is the number of elements in U Gosh. This cries out for an example.

31. 31 The Mandatory Example Let U = {a,b,c,d} Then P U = { , {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, (a,b,c} {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d} } Count them, there are 2 4 = 16!

32. 32 This concludes The Essence of Sets Tune in next time for more sets - why do we care? Fine print: The over-achieving student will now work the problem exercises.