Review of Sets and Set Operations Chapter 1-1 Review of Sets and Set Operations

What’s a Set? A set is a collection whose members are specified by a list or rule In the list, only have each element listed once (no need to have a duplicate element in a set) Ex: 𝑆= 𝐴𝑙𝑎𝑏𝑎𝑚𝑎, 𝐴𝑙𝑎𝑠𝑘𝑎, 𝐴𝑟𝑖𝑧𝑜𝑛𝑎, 𝐴𝑟𝑘𝑎𝑛𝑠𝑎𝑠 Set S is the set of states starting with the letter A When a rule is used to specify a set you will see it written as: 𝑆= 𝑥:… This is read “S is a set of all x such that… Nothing can be partially in a set. It is either in the set, or it isn’t. Finite sets have a finite number of elements within it Ex: 𝑌= 𝑦:𝑦 𝑖𝑠 𝑎𝑙𝑙 𝑒𝑣𝑒𝑛 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10 Infinite sets have infinite number of elements within it Ex: Z= 𝑧:𝑧 𝑖𝑠 𝑎𝑙𝑙 𝑜𝑑𝑑 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

Elements The set I of even positive integers less than 15 can be written a couple ways Ex: 𝐼= 𝑛:𝑛 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 15 Ex: 𝐼= 2,4,6,8,10,12,14 To indicate x is an element of a set X, we write 𝑥∈𝑋 To indicate x is not an element of set X, we write 𝑥∈𝑋 Ex: 2∈𝐼 “2 is an element of set I” **important note: lowercase letters indicate the element, uppercase letters indicate the set**

Subsets A set A is a subset of a set B, written 𝐴 𝐵, if every element in A is also in B. If A B and B A, then B and A have exactly the same elements, in which case we say 𝐴=𝐵 Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14 𝐼= 2,4,6,8,10,12,14 Since all elements in A are also in I, A I Since all elements in B are also in I, B I Is A B? Is B A?

Unions A Union is a combination of all the elements of 2 sets Let A and B be sets. The set 𝐴∪𝐵, called the union of A and B, consists of all elements which are in A or B or both. 𝐴∪𝐵= 𝑥:𝑥∈𝐴 𝑜𝑟 𝑥∈𝐵 Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14 𝐴∪𝐵= 2,4,6,8,10,12,14 Even though 2 is in both sets, we do not need to write 2 twice in the Union set

Intersections Intersections are like Unions, but opposite. An intersection of sets contains only elements that are in common in both sets Let A and B be sets. The set 𝐴∩𝐵, is called the intersection of A and B, consists of all elements which are in both A and B 𝐴∩𝐵= 𝑥:𝑥∈𝐴 𝑎𝑛𝑑 𝑥∈𝐵 Ex: 𝐴= 2,4,8,12 𝐵= 2,6,10,12,14 𝐴∩𝐵= 2,12 2 and 12 are the only values in both sets

Try this Let sets S, E, C, and M be defined as 𝑆= CT, MA, MD, CA, CO, MI, MN 𝐸= CT,MA,MD 𝐶= CA,CO,CT 𝑀= MA,MD, MI,MN Name any subsets you see. 𝐸∩𝐶= 𝐸∩𝑀= 𝐸∪𝐶= 𝐸∪𝑀= 𝑀∪𝐶= 𝑀∩𝐶=

Empty Set The set which contains no elements is known as the empty set, and is denoted by ∅. By convention, the empty set is a subset of every set Since the empty set has no elements, we say 𝐴∩∅=∅ and A∪∅=𝐴 Two sets A and B are disjoint if 𝐴∩𝐵=∅ This means sets A and B have no elements in common

Parentheses Matter! Let 𝐴= 𝑎,𝑏,𝑐 𝐵= 𝑎,𝑐,𝑒 𝐶= 𝑎,𝑑 Ex: 𝐴∩𝐵∩𝐶= Ex: 𝐴∪𝐵∪𝐶= Notice:

Universal Set and Complements A set U is said to be a universal set for a problem if all sets being considered in the problem are subsets of U. Let A be subset of U, then A’ is the complement of A, meaning A’ contains all the elements that are not in A. A’ is said “A Prime” Ex: Let 𝑈= CA,CO,CT,IL,IN 𝑋= CA,CT,IL 𝑌= CO,CT, IN 𝑍= CO,IN 𝑋 ′ = 𝑌 ′ = 𝑍 ′ = 𝑌∩ 𝑍 ′ = 𝑋∩ 𝑍 ′ = 𝑍∩ 𝑌 ′ =

Cartesian Product The Cartesian Product of sets A and B, denoted 𝐴×𝐵, is the set of all ordered pairs 𝑎,𝑏 where 𝑎∈𝐴 and 𝑏∈𝐵 𝐴×𝐵= 𝑎,𝑏 :𝑎∈𝐴, 𝑏∈𝐵 Ex: A survey can be conducted by either mail (M) or phone (P) in one of three cities: Atlanta (A), Boston (B), or Cincinnati (C). You must first choose a method (M, P) then a city (A, B, C). Each possible survey can be denoted as an ordered pair. 𝑀,𝐴 , 𝑀,𝐵 , 𝑀,𝐶 , 𝑃,𝐴 , 𝑃,𝐵 ,(𝑃,𝐶)

Cartesian Product (You Try) Let 𝐴= 𝑎,𝑐,𝑒 𝐵= 𝑏,𝑑,𝑒 𝐶= 𝑏,𝑑 \ Find 𝐴×𝐶= Find 𝐵×𝐶= Find 𝐶×𝐶= The order of elements within the braces is not important but the order of the symbols within the parentheses is! (𝑥,𝑦) is very different from (𝑦,𝑥)

Order Matters! A football league consists of 4 teams: Aardvarks (A), Bisons (B), Coyotes (C), and Dingos (D). Each game can be denoted as an ordered pair in which the first entry denotes the home team. Games: 𝐺=