1. Section 2.2 – Subsets and Set Operations
Introduction to Sets
Section 2.2 – Subsets and Set Operations
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2. Definitions A set is a collection of objects.
Objects in the collection are called elements of the set.
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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.
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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.
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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.
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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}.
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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 {1, 2, 3, … } is the set of all counting numbers.
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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.
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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.
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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.
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11. Definition
The set with no elements is called the empty set or the null set and is designated with the symbol .
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12. Examples – empty set
The set of all pencils in your bookbag might 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.
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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…
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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}
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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}
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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}
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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.
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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.
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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}
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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.
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21. Number of Elements in Subset Subsets with that Number of Elements
If we start with the set {x, y, z}, let’s look at how many subsets we can form:
Number of Elements in Subset
Subsets with that Number of Elements 3
{x, y, z}
One subset 2
{x, y}, {x, z}, {y, z}
3 subsets 1
{x}, {y}, {z} Ø
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22. Example 1 - Subset
Find all subsets of A = { American Idol, Survivor }. The subsets are: { American Idol, Survivor} { American Idol } { Survivor } Ø
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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
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24. Try examples 3 and 4 with a partner!
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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!
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26. Example 1 - intersection
If A = {3, 4, 6, 8} and
B = { 1, 2, 3, 5, 6} then
A  B = {3, 6}
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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 = 
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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 } A ∩ B = {x | x < 7 }
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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
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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}.
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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 }
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