Introduction to Set Theory

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

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

18. 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

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

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

21. 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 red or blue or purple. A  B

22. Definition – complement of • A complement of set A is the set of all elements in the universal set that are not in A. • 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.

23. Venn Diagram – A’

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

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

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