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Set definitions Dr. Ali Hussein Muqaibel Ver. 5.1 Dr. Ali Muqaibel 1 Probability & Random Processes for Engineers
Outlines (Set definitions) Dr. Ali Muqaibel 2 Element, Set, Class Set properties: Countable, uncountable Finite, infinite Subset & proper subset Universal set, empty=null set Two sets: Disjoint=mutually exclusive.
Set Definition Set : Collection of objects elements Class of sets: set of sets Notation: Set ? , element ? Notation: ? ∈ ? or ? ∉ ? How to define a set? 1. Tabular {.}: Enumerate explicitly 6,7,8,9 . 2. Rule: {Integers between 5 and 10}. Set Definition Examples: Dr. Ali Muqaibel 3 ? = {2,4,6,8,10, ⋯ } ? = {? ∈ 𝑁 ȁ ? is even} 𝑁 = the set of natural numbers ? = the set of integers ? = the set of rational numbers ? = the set of real numbers elements ? 1 ? 2 Set ? Class ? ? ?
Set Definitions: Countable/ finite Countable set: elements can be put in one-to- one correspondence with natural numbers 1,2,3,,… Not countable= uncountable . Set ? is finite : empty or counting its elements terminates i.e. finite number of elements Not finite : infinite. Finite => countable Uncountable => infinite Example: Describe 𝑨, 𝑵, 𝑹, 𝒁 ? Countable, uncountable, finite, infinite. Dr. Ali Muqaibel 4 ? = {2,4,6, . . . } is countable. 𝑁, ?, & ? are countable. ? is uncountable. ? = {2,4,6,8,10, ⋯ } ? = {? ∈ 𝑁ȁ ? is even} 𝑁 = the set of natural numbers ? = the set of integers ? = the set of rational numbers ? = the set of real numbers 0 1 Set of all real numbers between 0 and 1
Universal Set & Null Set Universal set ( ?) : all encompassing set of objects under discussion. Example: Tossing two coins: ? = {𝐻𝐻, 𝐻?, ?𝐻, ??} Rolling a die: ? = {1,2,3,4,5,6} For any universal set with 𝑁 elements, there are 2 𝑁 possible subsets of ?. Example: rolling a die, The universal set is ? = 1,2,3,4,5,6 , 𝑁 = 6 , there are 2 𝑁 = 64 subset. Empty set =null set= 𝝓 has no elements Dr. Ali Muqaibel 5
Subset, Proper Subset, and Disjoint sets The symbol denotes subset and denotes proper subset ? ⊆ ?: Every element of ? is also an element in ? . Mathematically: ? ∈ ? ⇒ ? ∈ ? ? ⊂ ?: at least one element in ? is not in ? . Statement: The null set is a subset of all other sets. Disjoint = Mutually exclusive : no common elements. ? ∩ ? = 𝜙 Dr. Ali Muqaibel 6 B A ? 1 B A
Set Definitions :Exercise For the following sets ..specify: (tabular/rule defined) (finite/infinite) (countable/uncountable) Dr. Ali Muqaibel 7 ? = {1,3,5,7}, ? = {1,2,3, ⋯ }, ? = {0.5 < ? ≤ 8.5} ? = {0.0}, ? = {2,4,6,8,10,12,14}, ? = {−5.0 < 𝑓 ≤ 12.0} Few examples: A is countable and finite ? is uncountable infinite ? is tabular format ? ⊂ ? ? ⊂ ? ? ⊂ ? Next, we will do some operations ? ∩ ? = 𝜙 ? ≠ 𝜙
Set Operations Dr. Ali Hussein Muqaibel Ver. 5.1 Probability & Random Processes for Engineers Objective: Venn Diagram, Equality & Difference, union and intersection, complement, Algebra of sets, De Morgan s laws, Duality principles. Dr. Ali Muqaibel 1
Outlines (Set operations) Venn diagram, Equality, difference, union=sum, intersection=product, complement. Algebra of sets (commutative, distributive, associative) De Morgan s law Duality principle Dr. Ali Muqaibel 2
Set Operations Venn Diagram: sets are represented by closed-plane figures. The universal sets 𝑆 is represented by a rectangle.

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