Four Basic Concepts: Sets, Functions, Relations, and Binary Operations PDF

FOUR BASIC CONCEPTS: Sets, Functions, Relations and Binary Operations Learning Targets ▰ Define the four basic concepts. ▰ Perform operations on the four basic concepts. ▰ Explain mathematics as a language of sets, relations, functions, binary operations....

FOUR BASIC CONCEPTS: Sets, Functions, Relations and Binary Operations Learning Targets ▰ Define the four basic concepts. ▰ Perform operations on the four basic concepts. ▰ Explain mathematics as a language of sets, relations, functions, binary operations. 2 1 SETS 3 SETS ▰ Sets are organized collection of objects. ▰ It can be represented in set-builder form or roster form. ▰ Usually, sets are represented in curly braces {}, for example, A = {1,2,3,4} is a set. 4 SETS ▰ A set is well-defined if it possible to decide whether an object belongs to a given set or not. ▰ Sets are denoted by any capital letter of the English alphabet, while, the elements by small letters. 5 WAYS OF DESCRIBIING A SET ROSTER METHOD: A = {2,4,6,8,10…30} SET BUILDER NOTATION: A = {x:x is a positive even integer from 2 to 40} 6  TYPES OF SETS ▰ Empty Set – set with no elements. ▰ Unit set – sets with single element. ▰ Equal Sets – sets with equal elements. ▰ Equivalent Sets – sets with equal number of elements. ▰ Finite sets – set that contains elements that are countable. ▰ Infinite sets – set that contains elements that has no end or not countable 7 Exercise A={} empty set 8 Exercise A={2} unit set 9 Exercise A = { apple } unit set 10 Exercise S = {∅} empty set 11 Exercise D = {1,2,3,4,5} finite set 12 Exercise X = {n,i,g,h,t} Y = {t,h,i,n,g} equal sets 13 Exercise A = {1,2,3,4,5} B = {a,b,c,d,e} equivalent sets 14 Exercise A = {5,6,7,8,9} B = {9,8,7,6,5} equal sets 15  TYPES OF SETS ▰ Universal Set – is the set of all elements under discussion ▰ Subset – A subset B, A⊆B, means that every element of A is also an element of B. 16  Examples The set of all the letters in the English Alphabet U = {a,b,c,d,…,z} 17  Examples The set of all the vowels is a subset of the set of the letters in the English Alphabet A = {a,e,i,o,u} B = {a,b,c,d,…,z} A⊆B 18 OPERATIONS ON SETS ▰ Union of sets ▰ Intersection of sets ▰ Difference of sets ▰ Complement of a set 19 Union of sets Example: If A = {1, 2, 3}, and B = {4, 5}, then A∪B A ∪ B = {1, 2, 3, 4, 5} 20 Union of sets Example: If A = {1, 2, 3, 4}, and B = {4, 5}, then A∪B A ∪ B = {1, 2, 3, 4, 5} 21 Intersection of sets Example: If A = {1, 2, 3}, and B = {2, 3, 4}, then A∩B A ∩ B = {2, 3} 22 Intersection of sets Example: If A = {1, 2, 3}, and B = {4, 5, 6}, then A∩B A ∩ B = {∅} 23 Difference of sets Example: If A = {1, 2, 3}, and B = {2, 3, 4}, then A-B A - B = {1} 24 Difference of sets Example: If A = {1, 2, 3}, and B = {4, 5}, then A-B A - B = {1, 2, 3} 25 Difference of sets Example: If A = {1, 2, 3, 4}, and B = {2, 4, 6, 8}, then A-B A - B = {1, 3} 26 Complement of sets Example: If U = {a, e, i, o, u}, and A = {a, e}, then A’ A’ = {i, o, u} 27 Complement of sets Example: If U = {1, 2, 3, 4, 5}, and A = {2, 4}, then A’ A’ = {1, 3, 5} 28 29  Seatwork Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 3, 5, 7}; B = {2, 4, 6, 8}; C = {3, 6, 9, 0} Find: 1. A∪B 6. C’ 2. B∪C 7. B - C 3. A∩C 8. C - A 4. A’ 9. A’ ∪ B 30 5. B’ 10. B ∩ C’ 2 RELATIONS 31 RELATION Set of ordered pair x and y. 32 DOMAIN AND RANGE X-values = input, independent variable, domain Y-values = output, dependent variable, range 33 WAYS OF REPRESENTING RELATIONS ▰ Ordered Pair ▰ Arrow Diagram ▰ Table ▰ Graph 34 TYPES OF RELATIONS ▰ One to One ▰ Many to one ▰ One to many – NOT FUNCTION 35 3 FUNCTIONS 36  FUNCTION “a relation in which every input has exactly one output” ALL FUNCTIONS ARE RELATIONS, BUT NOT ALL RELATIONS ARE FUNCTIONS. 37 WAYS OF REPRESENTING RELATIONS ▰ Ordered Pair ▰ Arrow Diagram ▰ Table ▰ Graph 38  VERTICAL LINE TEST A graph represents a function if and only if each vertical line intersects the graph at most once. 39  4 BINARY OPERATIONS 40 Unary vs. Binary ▰ -2 ▰ 4-2 41  BINARY OPERATIONS ▰ two quantities ▰ Any operation that combines two values to create a new one. ▰ Common binary operations are: Addition, subtraction, multiplication, and division 42  PROPERTIES OF BINARY OPERATIONS ▰ Commutative Property a*b=b*a ▰ Associative Property a*(b*c) = (a*b)*c ▰ Identity Property e*a=a*e ▰ Distributive Property (a+b)(c+d) = ac + ad + bc + bd 43 Seatwork bit.ly/SW2-Midterm 44 THANK YOU! 45