Chapter 3.1 The Language of Sets (concepts) PDF

CHAPTER 3. MATHEMATICAL LANGUAGE AND SYMBOLS – Language of Sets Core Idea “Mathematics has its own symbols, syntax, and rules.” learning objectives 1. discuss the definitions of a set, subset, and proper subset; 2. determ...

CHAPTER 3. MATHEMATICAL LANGUAGE AND SYMBOLS – Language of Sets Core Idea “Mathematics has its own symbols, syntax, and rules.” learning objectives 1. discuss the definitions of a set, subset, and proper subset; 2. determine whether a set is in roster or set- builder notation; 3. find all possible subsets of a given set; 4. perform set operations; and 5. illustrate subset and universal set using Venn Diagram.  What is a set? Is a group of tall buildings a set? Is a collection of musical instruments a set? Is a collection of nice shoes a set? Is a collection of subjects in Grade 10 a set?  set not a set collection of group of tall buildings musical instruments collection of subjects collection of nice shoes in Grade 10 SET It is a collection of related and well-defined objects called elements (denoted by ∈). : = {;|0 < ; < 5} " = {1, 2, 3, 4, 5, 6, 7, 8, 9} : = {1, 2, 3, 4} 0 = {123, 425567, 8592} Historical background Georg Cantor (1845-1918) introduced the word set in 1879. UNIVERSAL SET It is collection of all elements of all the related sets, known as its subsets. The universal set is usually denoted by !. Symbol Set ℝ real numbers ℝ$ positive real number ℤ all integers ℤ$ positive integers ℚ rational numbers ℕ natural numbers WRITING A SET q Roster Notation is a way of listing the elements separated by a comma. Example: Let ! and " be sets. If ! is the set of all even whole numbers between 1 and 10, and " is the set of all odd whole numbers between 1 and 10 write set ! and " in roster notation. Solution: ! = {2, 4, 6, 8} ; " = {3, 5, 7, 9} WRITING A SET q Set-Builder Notation it is a way of representing or explaining the properties that must satisfy by the elements of a set. Example: Consider ! and " as sets of natural numbers. Let ! = {1, 2, 4, 5, 6, 7} and " = 11,12,13,14. Write the sets in set-builder notation. Solution: ! = {. ∈ ℕ|0 <. < 8} " = {5 ∈ ℕ|10 < 5 < 15}  It is a set which contains no elements. It is EMPTY SET usually denoted as { } or ∅. The empty set is always considered a subset of any set. Is {0} a null set? No. CARDINALITY OF A SET Let " be a set. The cardinality of a set, denoted by -(") is the number of elements in ". Let " = {2,4,6,8,10}, then -(") = 5. SUBSET These are sets contained in a universal set or another set. Definition. If ! and " are sets, then ! is called a subset of ", denoted by ! ⊆ ", if and only if, every element of ! is also an element of ". Symbolically, ! ⊆ " means that for all elements $ ∈ !, then x ∈ ". ! ⊆ " is read as “!” is a subset of “"”. ! ⊈ " is read as “!” is not a subset of “"”. SUBSET These are sets contained in a universal set or another set. Example: Find the subsets of the following sets. a. P = #, %, & b. Q = #, (, ), * c. S = ,, -, (, ),. SUBSET These are sets contained in a universal set or another set. Solution: Find the subsets of the following sets. a. P = %, ', ( , % , ' , ( , %, ' , %, ( , ', ( , {%, ', (} , % , + , , , - , %, + , %, , , %, - , +, , , b. Q = %, +, ,, - +, - , ,, - , %, +, , , %, +, - , %, ,, - , +, ,, - , {%, +, ,, -} c. S = /, 0, +, ,, 1 How many subsets can be obtained here without enumeration? NUMBER OF SUBSETS How many subsets are there if the sets has: 1 element = 2 subsets Thus, to get the number 2 elements = 4 subsets of subsets of a given set, use the formula !". 3 elements = 8 subsets For example, if " # = % 4 elements = 16 subsets elements, then !% = %& subsets.  Definition. PROPER SUBSET Let ! and " be sets. ! is a proper is any subset of a set subset of ", if and only if, every except itself. element of ! is in " but there is at least one element of " that is not in !. ! ⊂ " is read as “!” is a proper subset of “"”. Example: Let ! = 1,4,3 ; " = 1,2,3,4,5 ; , = {2,3,1,4,5}. Is ! ⊂ "? YES. Is , ⊂ "? No. Is C ⊆ "? YES. POWER OF A SET It is the set of all subsets for any given set which includes the empty set. Example: Consider ! = {$, &, '}. Let ((!) denotes the power of a set !. So, P( S ) = { f , {a}, {r}, {} i , {a, i}, {a, r}, {i, r}, {i, r , a} } Based on ! and ((!) tell whether each of the following is TRUE or FALSE. a Î P(S ) {a}Î P(S ) {{a}} Ì P(S ) {a, r} Ì P( S ) End of Discussion…