Mathematics 07 - Language of Sets (November 13, 2024) PDF

OPENING PRAYER  MENTAL COMPUTATIONS Language of Sets November 13, 2024  Sets Definition. A Set is any well defined collection of definite and distinct “objects.” Definition. The elements of a set are the objects in a set. Notation. Usually we denote sets with upper-case letters,...

OPENING PRAYER  MENTAL COMPUTATIONS Language of Sets November 13, 2024  Sets Definition. A Set is any well defined collection of definite and distinct “objects.” Definition. The elements of a set are the objects in a set. Notation. Usually we denote sets with upper-case letters, elements with lower-case letters. The following notation is used to show set membership means that x is an element of the set A means that x is not an element of the set A.  Ways of Describing Sets Roster Method. Elements are listed. Rule Method. Elements in the set are described. A ={integers from 1 to 6} Set Builder Notation of the Rule Method  Practice Describe the following sets using roster method. {counting numbers less than 9} {vowels of the word “family”} {days of the week beginning with letter T} {letters of the word “mathematics”}  Practice Describe the following sets by rule method using set builder notation. {Saturday, Sunday} {a, e, i, o, u} {f, l, o, w} {0, 1, 2, 3, 4}  Kinds of Sets Null Set or Empty Set. This is a set with no elements. It is denoted by {} or ϕ. Singleton Set or Unit Set. This is a set with only one element. Finite Set. A set with element(s) than can be possibly listed or enumerated. Infinite Set. A set with elements that cannot be possibly listed or enumerated.  Kinds of Sets Universal Set. This is a set containing all objects or elements and of which all other sets are subsets.  Other Special Sets  Other Special Sets Joint Sets. Sets having common element(s). Disjoint Sets. Sets with no common element(s).  Special Sets of Numbers N = The set of natural numbers. {1, 2, 3, …} W = The set of whole numbers. {0, 1, 2, 3, …} Z = The set of integers. { …, -3, -2, -1, 0, 1, 2, 3, …} Q = The set of rational numbers. ={x| x=p/q, where p and q are elements of Z and q≠0} Q’ = The set of irrational numbers. R = The set of real numbers. C = The set of complex numbers.  Membership Relationships Definition. Subset. “A is a subset of B” We say “A is a subset of B” if , i.e., all the members of A are also members of B. The notation for subset is very similar to the notation for “less than or equal to,” and means, in terms of the sets, “included in or equal to.”  Membership Relationships Definition. Proper Subset. “A is a proper subset of B” We say “A is a proper subset of B” if all the members of A are also members of B, but in addition there exists at least one element c such that but. The notation for subset is very similar to the notation for “less than,” and means, in terms of the sets, “included in but not equal to.”  Cardinality of a Set If A is a set, the number of elements in A is called the cardinality of A and will be denoted by n(A).  Properties of Sets Every set is a subset of itself. Empty set is a subset of any set. For any two sets A and B, if A is a subset of B and B is a subset of A, then A = B.  Power Set Language of Sets Set Operations Union of Sets  Intersection of Sets “A intersects B” is the set of all elements that are in both A and B. This is similar to the logical “and”. A ∩ B = {x| x is in A and x is in B} Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5} Then A ∩ B = {1, 3, 5}. Note that 1, 3, 5 are in both A and B.  Set Complement or “A complement,” or “not A” is the set of all elements not in A. A’ = {x| x is not in set A}. Example: U={1,2,3,4,5}, A={1,2}, then A’ = {3,4,5}.  Difference of Sets The set difference “A minus B” is the set of elements that are in A, with those that are in B subtracted out. It is the set of elements that are in A, and not in B. Examples  Venn Diagrams Venn Diagrams use topological areas to stand for sets. I’ve done this one for you.  Venn Diagrams Try this one!  Venn Diagrams Here is another one Some Test Questions Some Test Questions Some Test Questions Some Test Questions Some Test Questions Some Test Questions Some Test Questions Some Test Questions  ACTIVITY: Find a pair and answer the practice exercises from the book on page 226 (Math Engage); and on page 239 (Written Math B: #36, #37, #39, #42, and #44 only)  ASSIGNMENT: Answer the Math Engage on page 240 (#70 and #71 only) CLOSING PRAYER