Section 2.2 – Subsets and Set Operations Introduction to Sets Section 2.2 – Subsets and Set Operations Applied Math

Definitions A set is a collection of objects. Objects in the collection are called elements of the set. Applied Math

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. Applied Math

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. Applied Math

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. Applied Math

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}. Applied Math

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. Applied Math

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. Applied Math

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. Applied Math

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. Applied Math

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

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. Applied Math

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… Applied Math

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} Applied Math

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} Applied Math

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} Applied Math

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. Applied Math

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. Applied Math

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} Applied Math

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. Applied Math

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 Applied Math

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

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 Applied Math

Try examples 3 and 4 with a partner! Applied Math

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! Applied Math

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

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 =  Applied Math

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 } Applied Math

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 Applied Math

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

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 } Applied Math