# CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

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1 CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences Set relations: Equal sets Subsets, proper subsets Disjoint sets =, Set operations Intersection Union Difference Complement ~, c Cartesian product x Venn diagrams Set Identities Partitions Power sets Proof problems on sets 1

2 1. Introduction Set: any collection of objects (individuals) Naming sets:,, C,. Members of a set: the objects in the set Naming objects: a, b, c,. Notation: Let be a set of 3 letters a, b, c. We write = {a, b, c} a is a member of, a is in, we write a є d is not a member of, we write d Important: 1. {a} a {a} - a set consisting of one element a. a - the element itself 2. set can be a member of another set: = { 1, 2, {1}, {2}, {1,2}} Finite sets: finite number of elements Infinite sets: infinite number of elements Cardinality of a finite set : the number of elements in : #, or Describing sets: a. by enumerating the elements of : for finite sets: {red, blue, yellow}, {1,2,3,4,5,6,7,8,9,0} for infinite sets we write: {1,2,3,4,5,.} b. by property, using predicate logic notation Let P(x) is a property, D - universe of discourse The set of all objects in D, for which P(x) is true, is : = {x P(x)} we read: consists of all objects x in D such that P(x) is true c. by recursive definition, e.g. sequences 2

3 Examples: 1. The set of the days of the week: {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} 2. The set of all even numbers : { x even(x) } { 2,4,6,8,.} 3. The set of all even numbers, greater than 100: { x even(x) x > 100} { 102, 104, 106, 108,.} 4. The set of integers defined as follows: a 1 = 1, a n+1 = a n + 2 (the odd natural numbers) Universal set: U - the set of all objects under consideration Empty set: Ø set without elements. 2. Relations between sets 2.1. Equality Let and be two sets. We say that is equal to, = if and only if they have the same members. Example: = {2,4,6}, = {2,4,6} = = {a, b, c}, = {c, a, b} = = {1,2,3}, = {1,3,5}, Written in predicate notation: = if and only if x, x є x є 3

4 2.2. Subsets The set of all numbers contains the set of all positive numbers. We say that the set of all positive numbers is a subset of the set of all numbers. Definition: is a subset of if all elements of are in. However may contain elements that are not in Notation: Formal definition: if and only if x, x є x є Example: = {2,4,6}, = {1,2,3,4,5,6}, Definition: if is a subset of, is called a superset of. Other definitions and properties: a. If and then = If is a subset of, and is a subset of, and are equal. b. Proper subsets: is a proper subset of,, if and only if is a subset of and there is at least one element in that is not in. iff x, x є x є x, x є x The above expression reads: is a proper subset of if and only if all x in are also in and there is an element x in that is not in. iff means if and only if The empty set Ø is a subset of all sets. ll sets are subsets of the universal set U. 4

5 2.3. Disjoint sets Definition: Two sets and are disjoint if and only if they have no common elements and are disjoint if and only if ~ x, (x є ) (x є ) i.e. x, x V x If two sets are not disjoint they have common elements. Picturing sets: Venn diagrams - used to represent relations between sets is a (proper) subset of ll elements in the set are also elements in the set Disjoint sets Not disjoint sets 5

6 3. Operations on sets 3.1. Intersections The set of all students at Simpson and the set of all majors in CS have some elements in common - the set of all students in Simpson that are majoring in CS. This set is formed as the intersection of all students in CS and all students at Simpson. Definition: Let and are two sets. The set of all elements common to and is called the intersection of and Notation: Formal definition: Venn diagram: = {x (x є ) (x є )} Example: = {2,4,6}, = {1,2,5,6}, = {2,6} Other properties:, The intersection of two sets and is a subset of, and a subset of Ø = Ø The intersection of any set with the empty set is the empty set U = The intersection of any set with the universal set is the set itself. Intersection corresponds to conjunction in logic. Let = {x P(x)}, = {x Q(x)} = {x P(x) Q(x)} 6

7 3. 2. Unions The set of all rational numbers and the set of all irrational numbers taken together form the set of all real numbers - as a union of the rational and irrational numbers. ll classes at Simpson consist of students. If we take the elements of all classes, we will get all students - as the union of all classes. Definition: The union of two sets and consists of all elements that are in combined with all elements that are in. (note that an element may belong both to and ) Notation: Formal definition: Venn diagram: = {x (x є ) V (x є )} E Example: = {2,4,6,8,10}, = {1,2,3,4,5,6}, = { 1,2,3,4,5,6,8,10} contains all elements in and without repetitions. Other properties of unions: is a subset of the union of and, is a subset of the union of and Ø = The union of any set with the empty set is U = U The union of any set with the universal set E is the universal set. Union corresponds to disjunction in logic. Let = {x P(x)}, = {x Q(x)} = {x P(x) V Q(x)} 7

8 3.3. Differences The books I have at home are the books that are mine and the books I've taken from the library. When I return the books to the library, only my own books will remain. Thus my own books at home form a set that is the difference between all books at home and library books Definition: Let and be two sets. The set -, called the difference between and, is the set of all elements that are in and are not in. Notation: or \ Formal definition: Venn diagram: = { x ( x є ) ( x )} Example: = {2,4,6}, = {1, 5,6}, - = {2,4} Ø = U = Ø The difference between and the empty set is The difference between and the universal set is the empty set Complements Definition: Let be a set. The set of all objects within the universal set that are not in, is called the complement of. Notation: ~ Formal definition: Venn diagram: ~ = {x x } ~ 8

9 Problems on set operations 1. Let = {1,3,4,5}, = {3,5,7,8} = = = Let E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 0} ~ = ~ = 2. Let the universal set be the set R of all real numbers and let = {x R -5 x < 3}, = { x R -1 < x 9}. Find each of the following: = = ~ = ~ = ~ ~ = ~ ~ = ~( ) = ~( ) = 3. Let E be the set of all student in Simpson College. : all CS majors : all Math majors C: all female students D: all senior students. Write the expressions for: a. ll female students that are both CS and Math majors b. ll male students and all Math senior students. c. ll CS students that are not senior students 9

10 Venn diagrams 4. Let, and C are sets represented below: Shade the regions corresponding to: ( ) C ( C) - ( C) ~( ) 10

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