Section 3.1 Sets and their operation

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.” 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: 1.{Doug, Amy, John, Jessica} 2.{TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF} 3.{ {A,B}, {A,C}, {B,C} } 4.{ }

Common sets of numbers Page 182 N … set of natural numbers {0, 1, 2, …} Z … set of integers {…, -2, -1, 0, 1, 2, …} Q … set of rational numbers R … set of real numbers

Definitions If A and B are sets, then the notation A  B (read “A is a subset of B”) means that every element of set A is also an element of set B. Practice. Which is true? 1.{1, 2, 3, 4}  {2, 3, 4} 2.Z  Q 3.Z  N 4.{ }  {a, b, c} 5.{3, 5, 7}  {2, 3, 5, 7, 11} 6.{a, b}  { {a, b}, {a, c}, {b, c} } 7. {a}  { {a, b}, {a, c}, {a, b, c} }

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: 1.(Property) {n  Z : n is divisible by 4} 2.(Form){4k : k  Z}

Practice with property description List five members of each of the following sets: 1.{n  N : n is an even perfect square } 2.{x  Z : x – 1 is divisible by 3 } 3.{r  Q : r 2 < 2 } 4.{x  R : sin(x) = 0 }

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

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 A and B. A  B (read “A union B”) is the set of elements in either set A or B. A – B (read “A minus B”) is the set of elements in set A which are not in B. A’ (read “the complement of A”) is the set of elements in the universe U which 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: 1.A  C 2.B  D 3.B – D 4.B’ 5.(A  B) – C 6.(A  C)  B 7.B’  C’ 8.(B  C)’ 9.(C  D) – A 10.B  D’

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) = – 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)