Warm Up September 19th/20th

Warm Up September 19th/20th • In a class of 50 students, 18 take Chorus, 26 take Band, and 2 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band? • In a school of 320 students, 85 students are in the band, 200 students are on sports teams, and 60 students participate in both activities. How many students are involved in either band or sports?

Basic Set Concepts Sets and their operation

Notes: Definitions A set S is collection of objects. These objects are said to be members or elements of the set, and the shorthand for writing “x is an element of S” is “x S.” Capital letters are used to name sets and the elements of the set are enclosed in braces.

There are three methods of representing the elements of a set: • [a] word description • [b] roster method • [c] set-builder notation

[a] word description • This is where the elements are described in words • Examples • (Set N is the set of natural numbers) • (Set Z is the set of integers) • (Set Q is the set of rational numbers) • (Set R is the set of real numbers)

[b] roster method The easiest way to describe a set is by simply listing its elements, the “roster method”. For example, the collection of odd one-digit numbers could be written {1, 3, 5, 7, 9}. Note that this is the same as the set {9, 7, 5, 3, 1} since the order elements are listed does not matter in a set.

Examples The elements of a set do not have to be numbers as the following examples show: • {Doug, Amy, John, Jessica} • {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF} • { {A,B}, {A,D}, {B,D} } • { }

c] set-builder notation This is mathematical notation for describing a set by stating the properties that its members must satisfy. ...

For instance, Natural Numbers… • N = {xx  Z, and x > 0} • We read this as “N is the set of all x such that x is an element of all integers and x is greater than zero”.

On another note: • The symbol  is used to indicate that an object is a member of a set • The symbol  indicates that an object is not an element of a set. • Examples: V = { p,q,r x} • x  V • m  V

The empty set or null set is the set that contains no elements and is represented by { } or Φ; it is a subset of every set.

Just a few more things to note: • A set is finite if the number of elements of the set can be counted (0 elements, 5 elements, etc.). • It is infinite if the number of elements continues to infinity, like the Natural numbers: N = {1, 2, 3, …}. • The three dots indicates that the elements continue in the same pattern.

Practice : Let’s use all three types of set notation to show the set of whole numbers, W. Word Description: (Set W is the set of whole numbers) Roster Method: W = {0,1,2,3,…} Set-Builder Notation W = {x l x  Z and x ≥ 0}

Common sets of numbers • W … set of whole numbers {0,1,2,3,…} • N … set of natural numbers {1, 2, …} • Z … set of integers {…, -2, -1, 0, 1, 2, …} • Q … set of rational numbers • R … set of real numbers

Notes: Definitions • The cardinal number of a set A where n(A) is the number of elements in set A. • Sets are equal if they contain exactly the same elements regardless of order. • Sets are equivalent if they contain the same number of elements; that is, n(A) = n(B). • Repeating elements in a set does not add a new element to the set.

Now you Practice: Show whether 5 is a member of W. Show whether –3 is a member of W. Use word notation to describe X = {3, 6, 9,…}

Practice: Use roster notation to describe the set of even Natural Numbers that are less than 10. Use set builder notation to describe odd Natural Numbers greater than 100. If E = {2, 4, 6, …} and O = {1, 3, 5, …} and F = {5, 10, 15, …}, then 5 ___ E, 30 __ F, and 17,385 ___ O.

Try These: • Find the cardinal number of each of the following sets: • A = {6, 10, 10, 14, 15, 16} • B = {872} • C = {9, 10, 11, …, 15, 16 • D = { } • Given the sets X = {0, 1, 1, 2, 3, 5, 5, 8}, Y = {1, 5, 3, 0, 8, 2}, and Z = {7, 8, 9, …, 12}, which are equal and which are equivalent?

Venn Diagrams and Subsets

Definitions The Universal Set is a set that contains all the elements under discussion, such as the set of all dogs or all musicians or all real numbers. If A and B are sets, then the notation AB (read “A is a subset of B”) means that every element of A is also an element of B.

Set theory definitions con’t: • Set A is a propersubset of B, A  B, if every element of A is also in B but A  B. • A  B indicates that A is not a proper subset of B. • The null set  or empty set { } is a proper subset of every set except itself. • The number of subsets of a set with n elements is 2n. • The number of proper subsets of a set with n elements is 2n - 1

Subset practice: Practice. Which is true? • {1, 2, 3, 4}  {2, 3, 4} • ZQ • ZN • { }  {a, b, c} • {3, 5, 7}  {2, 3, 5, 7, 11} • {a, b}  { {a, b}, {a, c}, {b, c} }

Practice of Proper Subsets: • If V = {x, y, z}, how many subsets and proper subsets can be found? • List all possible subsets of V, above. • Show whether N, the set of natural numbers, is a proper subset of W. • 3. Show whether the set of negative numbers, M, are a subset of W.

Define a universal set for D = {collies, poodles, terriers}, C = {lions, tigers, leopards}. • Define a universal set of D and C above plus A = {monkeys, birds, alligators}. • Given the set A = {lions, tigers, bears, and wolves} determine whether the following statements are true or false: [a] bears  A; [b] bears  A; [c] {bears}  A, [d] Φ  A; [e] Φ  A.

MORE notation! • Complement: • The complement of set A, A’, is the set of elements that are not in set A but are in the universal set; • That is, A’ = {xx  U and x  A}. • If U = {0, 1, 2, …} and B = {0, 1, 2, … 100}, the complement of B is shown as • B’ = {101, 102, 103, …}.

Try these: • If J = {all months beginning with J}, find J’. • If the universal set is all integers and N = {1, 2, 3, …} , find N’.

Homework: • Practice Worksheet

Definitions of set operations Let A and B be sets with elements from a specified universal set U. • AB (read “A intersect B”) is the set of elements in both sets Aand B. • A B (read “A union B”) is the set of elements in either set Aor B. • A – B (read “A minus B”) is the set of elements in set Awhich are not in B. • A’ (read “the complement of A”) is the set of elements in the universe Uwhich are not in A.

Practice with set operations Let A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}, C = {2, 3, 5, 7}, D = {6, 7, 8, 9, 10} be sets with elements from the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Find each of the following: • A  C • B  D • B – D • B’ • (A  B) – C • (A  C)  B • B’  C’ • (B  C)’ • (C  D) – A • B  D’

Set scavenger hunt • See worksheet for this in-class activity

Venn diagrams

Inclusion-Exclusion Principle The notation n(A) means “the number of elements of A.” For example, if A = {2, 3, 6, 8, 9}, then n(A) = 5. Principle of Inclusion/Exclusion for two sets A and B: n(AB) = n(A) + n(B) – n(AB)

Inclusion-Exclusion Principle Example. A = { 2, 4, 6, 8, …, 96, 98, 100 } and B = { 5, 10, 15, 20, …, 90, 95, 100} n(AB) = n(A) + n(B) – n(AB) = 50 + 20 – 10 = 60

Inclusion-Exclusion Principle Principle of Inclusion/Exclusion for three sets A, B, and C: n(AB C) = n(A) + n(B) + n(C) – n(AB) – n(AC) – n(BC) + n(ABC)

Exercises from Section 3.1 1-3, 5-7, 9-11, 15(a-d), 16(a-d), 17(b-e), 18, 26, 28, 29

Set notation Large sets cannot be listed in this way so we need the more compact “set-builder” notation. This comes in two types exemplified by the following: • (Property) {nZ : n is divisible by 4} • (Form) {4k : kZ}

Practice with property description List five members of each of the following sets: • {nN : n is an even perfect square } • {xZ : x – 1 is divisible by 3 } • {rQ : r2 < 2 } • {xR : sin(x) = 0 }

Practice with form description List five members of each of the following sets: • { 3n2 : nZ } • { 4k + 1 : kN } • { 3 – 2r : rQ and 0 r 5 }

Warm Up September 19th/20th