1. August 2006 Copyright © 2006 by DrDelMath.Com 1 Introduction to Sets Basic, Essential, and Important Properties of Sets

2. August 2006 Copyright © 2006 by DrDelMath.Com 2 Definitions A set is a collection of objects. A set is a collection of objects. Objects in the collection are called elements of the set. Objects in the collection are called elements of the set.

3. August 2006 Copyright © 2006 by DrDelMath.Com 3 Examples - set The collection of persons living in Rochester is a set. The collection of persons living in Rochester is a set. Each person living in Rochester is an element of the set. Each person living in Rochester is an element of the set. The collection of all counties in the state of New York is a set. The collection of all counties in the state of New York is a set. Each county in New York is an element of the set. Each county in New York is an element of the set.

4. August 2006 Copyright © 2006 by DrDelMath.Com 4 Notation Sets are usually designated with capital letters. Sets are usually designated with capital letters. Elements of a set are usually designated with lower case letters. Elements of a set are usually designated with lower case letters. We might talk of the set B. An individual We might talk of the set B. An individual element of B might then be designated by b. element of B might then be designated by b.

5. August 2006 Copyright © 2006 by DrDelMath.Com 5 Notation The roster method of specifying a set consists of surrounding the collection of elements with braces. The roster method of specifying a set consists of surrounding the collection of elements with braces. Think of a roster as just a comma-separated list enclosed in curly braces. Think of a roster as just a comma-separated list enclosed in curly braces.

6. August 2006 Copyright © 2006 by DrDelMath.Com 6 Example – roster method For example the set of counting numbers from 1 to 5 would be written as For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}. {1, 2, 3, 4, 5}.

7. August 2006 Copyright © 2006 by DrDelMath.Com 7 Example – roster method A variation of the simple roster method uses the ellipsis ( … ) when the pattern is obvious and the set is large. A variation of the simple roster method uses the ellipsis ( … ) when the pattern is obvious and the set is large.ellipsis {1, 3, 5, 7, …, 9007} is the set of odd counting numbers less than or equal to 9007. {1, 2, 3, … } is the set of all counting numbers.

8. August 2006 Copyright © 2006 by DrDelMath.Com 8 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

9. August 2006 Copyright © 2006 by DrDelMath.Com 9 Venn Diagrams To learn a bit more about Venn diagrams and the man John Venn who first presented these diagrams click on the history icon at the right. History

10. August 2006 Copyright © 2006 by DrDelMath.Com 10 Venn Diagrams Venn Diagrams are used in mathematics, logic, theological ethics, genetics, study theological ethicsgeneticsstudytheological ethicsgeneticsstudy of Hamletof Hamlet, linguistics, reasoning, and linguisticsreasoning of Hamletlinguisticsreasoning many other areas.

11. August 2006 Copyright © 2006 by DrDelMath.Com 11 Definition The set with no elements is called the empty set or the null set and is designated with the symbol . The set with no elements is called the empty set or the null set and is designated with the symbol .

12. August 2006 Copyright © 2006 by DrDelMath.Com 12 Examples – empty set The set of all pencils in your briefcase might indeed be the empty set. The set of all pencils in your briefcase might indeed be the empty set. The set of even prime numbers greater than 2 is the empty set. The set of even prime numbers greater than 2 is the empty set.

13. August 2006 Copyright © 2006 by DrDelMath.Com 13 Definition - subset The set A is a subset of the set B if every element of A is an element of B. The set A is a subset of the set B if every element of A is an element of B.Example: B = {oranges, grapes, apples, mangoes} A = {grapes, apples} A is a subset of B

14. August 2006 Copyright © 2006 by DrDelMath.Com 14 Example - subset The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7, 9} because 3 is an element of A but is not an element of B. The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7, 9} 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. The empty set is a subset of every set, because every element of the empty set is an element of every other set.

15. August 2006 Copyright © 2006 by DrDelMath.Com 15 Definition - intersection The intersection of two sets A and B is the set containing those elements which are The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B. elements of A and elements of B. We write A  B

16. August 2006 Copyright © 2006 by DrDelMath.Com 16 Example - intersection If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then B = { 1, 2, 3, 5, 6} then A  B = {3, 6} A  B = {3, 6}

17. August 2006 Copyright © 2006 by DrDelMath.Com 17 Example - intersection If A = { , , , , , , , ,  } and B = { , , , @, ,  } then A ∩ B = { ,  } If A = { , , , , , , , , } and B = { , , ,  } then A ∩ B = { , , ,  } = B

18. August 2006 Copyright © 2006 by DrDelMath.Com 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 portion of A, the purple colored region illustrates the intersection A ∩ B of A and B Excellent online interactive demonstration demonstration

19. August 2006 Copyright © 2006 by DrDelMath.Com 19 Definition - union The union of two sets A and B is the set containing those elements which are The union of two sets A and B is the set containing those elements which are elements of A or elements of B. elements of A or elements of B. We write A  B

20. August 2006 Copyright © 2006 by DrDelMath.Com 20 Example - Union If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then B = { 1, 2, 3, 5, 6} then A  B = {1, 2, 3, 4, 5, 6}. A  B = {1, 2, 3, 4, 5, 6}.

21. August 2006 Copyright © 2006 by DrDelMath.Com 21 Example - Union If A = { , , , , ,  } and B = { , , , @, ,  } then A  B = { , , , ,, , , , @,  } If A = { , , , ,  } and B = { , ,  } then A  B = { , , , ,  } = A

22. August 2006 Copyright © 2006 by DrDelMath.Com 22 Venn Diagram - union A is represented by the red circle and B is represented by the blue circle. The purple colored region illustrates the intersection. The union consists of all points which are colored red or blue or purple. Excellent online interactive demonstration demonstration A  B A∩BA∩B