Introduction to Sets: Definitions and Operations

1. Unit 4, Section 1 – Subsets and Set Operations Introduction to Sets

2. Definitions • 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 Monmouth County is a set. • Each person living in Monmouth is an element of the set. The collection of all counties in the state of New Jersey is a set. • Each county in New Jersey 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 bookbag is a set. • Each pencil in your bookbag is an element of the set.

5. Notation • Sets are usually designated with capital letters. • Elements of a set are usually designated with lower case letters. • We might talk of the set B. An individual element of B might then be designated by b.

6. Notation • The roster method of specifying a set consists of surrounding the collection of elements with braces. • For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.

7. Example – roster method A variation of the simple roster method uses the ellipsis ( … ) when the pattern is obvious and the set is large. {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. 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.

9. 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}. {x | x is a fraction whose numerator is 1 and whose denominator is a counting number }. Set builder notation will become much more concise and precise as more information is introduced.

10. 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.

11. Definition • The set with no elements is called the empty set or the null set and is designated with the symbol .

12. Examples – empty set The set of all pencils in your bookbagmight indeed be the empty set. The set of even prime numbers greater than 2 is the empty set. The set {x | x < 3 and x > 5} is the empty set.

13. Definition • The universal set, symbolized U, is the set of all potential elements under consideration. • Examples: All college students, all whole numbers, all the towns in a county…

14. Definition • The complement of a set A, symbolized A’,is the set of elements contained in the universal set that are not in A. • Using set-builder notation, the complement of A is A’ = {x | x  U and x  A}

15. Example 1 – Complement Let U = {v, w, x, y, z} and A = {x, y, z}. Find the complement A’. Solution: Cross out the elements in U that are also in A… so, A’ = {v, x}

16. Ex. 2 - Try this one on your own! Let U = {10, 20, 30, 40, 50, 60, 70 ,80, 90} and A = {10, 30, 50}. Find the complement A’. Solution: A’ = { 20, 40, 60, 70 , 80, 90}

17. Definition - subset • 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 and B contains elements which are not in A, then A is a proper subset of B.

18. Notation - subset 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.

19. Subsets 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}

20. Subsets 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. Notes about Sets… Every set is a subset of itself -> all elements in A are of course in A, so A  A. The empty set is a subset of every set, because every element of the empty set is an element of every other set.

21. Subsets If we start with the set {x, y, z}, let’s look at how many subsets we can form:

22. Example 1 - Subset Find all subsets of A = { American Idol, Survivor }. The subsets are: { American Idol, Survivor} { American Idol } { Survivor } Ø

23. Ex. 2 - Try this on your own! Find all subsets of A = { Verizon, AT&T, T-Mobile }. The subsets are: {Verizon, AT&T, T-Mobile } {Verizon, AT&T } { AT&T, T-Mobile } {Verizon, T-Mobile } {Verizon } { AT&T } {T-Mobile } Ø

24. Try examples 3 and 4 with a partner!

25. Definition - intersection • 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 • Think of street intersections, it is where BOTH roads (sets) meet!

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

27. Example 2 – Try on your own! 1. If A is the set of prime numbers and B is the set of even numbers then A ∩B = ??? Solution: A ∩B = { 2 } 2. If A = {x | x > 5 } and B = {x | x < 3 } then A ∩B = ??? Solution: A ∩B = 

28. Example 3 – More Problems to Try! If A = {x | x < 4 } and B = {x | x >1 } then A ∩B = ?? A ∩B = {x | 1 < x < 4 } If A = {x | x > 4 } and B = {x | x >7 } then A ∩B = ?? A ∩B = {x | x < 7 }

29. Definition - union • 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

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

31. Example 2 – Try These! 1. If A is the set of prime numbers and B is the set of even numbers then A B = ???? A B = {x | x is even or x is prime } 2. If A = {x | x > 5 } and B = {x | x < 3 } then A B = ???? A B = {x | x < 3 or x > 5 }