Math 141, Spring 2014,cBenjamin Aurispa6.1 Sets and Set Operations• A set is a collection of objects. The objects in a set are called the elements of the set.Usually sets are denoted by uppercase letters, A, B, C, . . ., and elements are usually denoted by low-ercase letters, a, b, c, . . ..There are two ways to write a set: roster notation and set-builder notation.• Roster notation is where the set is written by listing all the elements of a set inside braces s uch asA = {a, e, i, o, u}.• Set-builder notation is where the set is written in terms of a rule or property that describes all theelements in the set.Write the set A above in set-builder notation:• If a is an element of a set A, then we write a ∈ A. If a is not an element in A, then a /∈ A.If A = {1, 2, 3, 5, 7}, then 3 ∈ A but 4 /∈ A.• Two sets A and B are equal, A = B if and only if they have exactly the same elements. It doesn’tmatter what order they are written in.If A = {1, 3, 4, 7, 9} and B = {3, 7, 4, 9, 1}, then A = B.• We say a set A is a subset of B if every element of A is also an element of B. We denote this byA ⊆ B.If A = {1, 3, 4, 7, 9}, B = {3, 7, 4, 9, 1}, and C = {1, 4, 7}, list all subset relationships.• If A ⊆ B but A 6= B, then we say that A is a proper subset of B, and we denote this by A ⊂ B. Inother words, A is essentially a ‘smaller” subset of B. List all proper subset relationships from above.Make sure you know the difference between ∈ and ⊆. ∈ is u sed w hen we are talking ab ou t anELEMENT being in a set. ⊆ is used when we are talking about a SET being a subset of another set.Which of the following is a TRUE statement if A = {1, 2, 3, 4} and B = {2, 4}.2 ∈ A 2 ⊂ A {2} ⊆ A {4} ⊂ B {3} ∈ A B ⊂ A B ⊆ A B ⊂ B A ⊆ A1Math 141, Spring 2014,cBenjamin Aurispa• The empty set is the set that contains no elements and is den oted by ∅ or by {}.FACT: The empty set is a subset of EVERY set. For any set A, ∅ ⊆ A or {} ⊆ A.Example: List ALL subs ets of the set C = {1, 4, 7}.If a set A has n elements in it, then the total number of s ubsets of A is 2n.The number of proper subsets of A is .If A contains 4 elements, how many sub sets does A have?Proper subsets?• A universal set is the set of all elements of interest in a particular discus sion. It can vary dependingon the problem.• A Venn diagram is a visual representation of a set.Examples: Draw Venn diagrams to illustrate the following scenarios.A is a pr oper subset of B. (A ⊂ B). A and B have no elements in common.Set Operations• The union of two sets A and B, written A ∪ B, is the set of all elements that are IN A OR B ORBOTH. This is the analog to ∨, the inclus ive disjunction, in logic.2Math 141, Spring 2014,cBenjamin Aurispa• The intersection of two sets A and B, written A ∩ B, is the set of all elements that A and B havein common. In other words, it is the set of elements that are IN BOTH A AND B at the same time.This is the analog to ∧, the conjunction, in logic.If two sets A and B have no elements in common, then A ∩ B = ∅ and we say A and B are disjoint.• The complement of a set A, written Acis the set of all elements that are NOT IN A (but still in theuniversal set U of the p roblem). This is the analog to ∼, the negation, in logic.Example: Let U = {n, 2, 3, 4, w, 6, 7, 8, 9 }, A = {n, w, 7}, B = {x|x is an even number between 1 and 9},C = {n, 3, 4, 9}. Find the following sets.• A ∪ B• A ∩ C• Cc• A ∩ (B ∪ C)c• (Ac∩ C) ∪ Bc3Math 141, Spring 2014,cBenjamin AurispaSets: U = {n, 2, 3, 4, w, 6, 7, 8, 9}, A = {n, w, 7}, B = {x|x is an even number between 1 and 9},C = {n, 3, 4, 9}• (A ∪ B ∪ C)c• (A ∩ B ∩ C)cProperties of Set Operations1. Uc= ∅ and ∅c= U2. (Ac)c= A3. A ∪ Ac= U4. A ∩ Ac= ∅De Morgan’s Laws:• (A ∪ B)c= Ac∩ Bc• (A ∩ B)c= Ac∪ BcYou can think of these De Morgan’s Laws as a kind of distribu tive property for sets.Verify the first De Morgan Law using a Venn diagram.A BA BA BA B4Math 141, Spring 2014,cBenjamin AurispaConsider the following sets. Let U be the universal set of all undergraduate students at Texas A&M.M = {x ∈ U|x is a male}F = {x ∈ U|x is a freshman}S = {x ∈ U|x is a senior}Using these three sets, write the set that represents the following statements.The s et of students at A&M who are male freshman.The s et of students at A&M who are male seniors or female f reshmen.The s et of female students at A& M who are not seniors.The s et of students at A&M who are neither freshman nor seniors.What do the following sets represent in words?M ∪ FcM ∩ (F ∪ S)Consider the following 3 sets in the same universal set of undergraduate students at A&M.I = {x ∈ U|x has an iPod}D = {x ∈ U|x has a digital camera}L = {x ∈ U|x has a laptop}Using these three sets, write the set that represents the following statements.The s et of students at A&M who have an iPod and a digital camera but not a laptop.The s et of students at A&M who only have a digital camera.What does the set I ∪ D ∪ L represent in words?Shading Venn DiagramsShade the appropriate region in a 3-circle Venn diagram.A ∩ Bc∩ CcA BCabcdefgh5Math 141, Spring 2014,cBenjamin Aurispa(A ∩ C) ∪ (B ∩ Cc)A BCabcdefgh(Bc∪ C) ∩ (A ∪ Cc)A BCabcdefgh(Ac∪ B)c∩ CA BCabcdefgh6Math 141, Spring 2014,cBenjamin Aurispa6.2 The Number of Elements in a SetWe denote the number of elements in a set A as n(A).If A = {1, 6, a}, then n(A) =.If B = {x | x is a consonant in the alphabet }, then n(B) =n(∅) =Example: In a room of 100 students, 30 are juniors, 50 are female, an d 10 are female juniors. Fill in theVenn diagam below with the appropriate number of students in each region. Describe the students in regiona.baJ FcdCounting n(A ∪ B):A BA BUnion RuleIf A and B are any two finite sets, then the following formula holds.n(A ∪ …
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