1. Introduction to Set Theory

2. Introduction to Sets – the basics A set is a collection of objects. Objects in the collection are called elements of the set.

3. Examples - set The collection of persons living in Pembroke Pines is a set. ◦Each person living in Pembroke Pines is an element of the set. The collection of all counties in the state of Florida is a set. ◦Each county in Florida is an element of the set.

4. Examples - set The collection of counting numbers is a set. ◦Each counting number is an element of the set. The collection of pencils in your backpack is a set. ◦Each pencil in your backpack is an element of the set.

5. Notation Sets are usually designated with capital letters. ◦A = {a, e, i, o, u} Elements of a set are usually designated with lower case letters. ◦In the set A above, the individual elements of the set are designate as, a, e, i, etc.

6. Roster Method For example, the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}. For example, the set of animals found on a farm would be written as {chickens, cows, horses, goats, pigs }.

7. Notation Set builder notation has the general form {variable | descriptive statement }. The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

8. Example – set builder notation {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.

9. Notation – is an element of If x is an element of the set A, we write this as x  A. x  A means x is not an element of A. If A = {3, 17, 2 } then 3  A, 17  A, 2  A and 5  A. If A = { x | x is a prime number } then 5  A, and 6  A.

10. Notation – is a subset of The set A is a subset of the set B if every element of A is an element of B. If A is a subset of B we write A  B to designate that relationship. If A is not a subset of B we write A  B to designate that relationship.

11. Example - subset The set A = {1, 2, 3} is a subset of the set B ={1, 2, 3, 4, 5, 6} because each element of A is an element of B. We write A  B to designate this relationship between A and B. We could also write {1, 2, 3}  {1, 2, 3, 4, 5, 6}

12. Example – not a subset of The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7} because 3 is an element of A but is not an element of B. The empty set is a subset of every set, because every element of the empty set is an element of every other set.

13. Definition – universal set A universal set is a set which contains all objects under consideration, including itself, and is denoted as an uppercase U. For example, if we define U as the set of all living things on planet earth, the set of all dogs is a subset of U.

14. Venn Diagrams It is frequently very helpful to depict a set in the abstract as the points inside a circle (or any other closed shape). We can picture the set A as the points inside the circle shown here. A

15. Definition – universal set A universal set is a set which contains all objects, including itself. In a Venn Diagram, the Universal Set is depicted by putting a U box around the whole thing.

16. Operations on sets There are three basic set operations: - Intersection - Union - Complement

17. Definition – intersection of The intersection of two sets A and B is the set containing those elements which are and elements of A and elements of B. We write A  B

18. Example - intersection If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A  B = {3, 6}

19. Venn Diagram - intersection A is represented by the red circle and B is represented by the blue circle. When B is moved to overlap a A portion of A, the purple colored region B illustrates the intersection of A and B A ∩ B

20. Definition – union of The union of two sets A and B is the set containing those elements which are or elements of A or elements of B. We write A  B

21. Example - union If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6}, then A  B = {1, 2, 3, 4, 5, 6}.

22. Venn Diagram - union A is represented by the red circle and B is represented by the blue circle. A The purple colored region illustrates the intersection. B The union consists of all points which are colored oror red or blue or purple. A  B

23. Definition – complement of A complement of a set is the set of those elements which does not belong to a given set but belongs to the universal set. A complement is denoted by A’, A, or ~ A. The number of elements of A and the number of elements of A’ make up the total number of elements in U.

24. Venn Diagram – A’

25. Given the following Venn diagram, shade in (B UC)  ~ A.

26. First work with what is inside the parentheses. The union of B and C shades both circles fully. +

27. Then do the complement A by cutting out the overlap with A.

28. Definition – cross product The cross product of two sets A and B, denoted by A x B (read A cross B), is defined as the set of all ordered pairs (a,b) where a is a member of A and b is a member of B. For example: A = {1, 2, 3} B = {a,b} A x B = {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)} +9*696/8 +

29. Algebraic Properties Union and intersection are associative operations. (A  B)  C = A  (B  C) (A ∩ B) ∩ C = A ∩ (B ∩ C)

30. Algebraic Properties Union and intersection are commutative operations. A  B = B  A A ∩ B = B ∩ A

31. Algebraic Properties Two distributive laws are true. A ∩ ( B  C )= (A ∩ B)  (A ∩ C) A  ( B ∩ C )= (A  B) ∩ (A  C)