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Transcript

Chapter 1: Set Notation 1. Math 3201 Handout #1 Set a collection of distinguishable objects the set notation symbol is brackets { }; not square [ ] or round ( ) sets are equal if they contain the exact same elements even if they are in different orders a finite set is one which has a countable number of elements. Ex the set of integers between 0 and 20 an infinite set is one that has an infinite number of elements. Ex the set of real numbers sets can represented by: (i) using words or a sentence; (ii) listing the elements; (iii) using set notation ex. {the whole numbers less than 10}; {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; {x / x 10, x W } 2. Element each object in a set the set notation symbol for "is an element of" is ∈ ex. 2 is an element of the set of whole numbers: 2 ∈ 𝑊 3. Number of Elements the number of objects in a set denoted by 𝑛(𝐴) ex. If A = {2, 4, 6, 8, 10, 12} then 𝑛(𝐴) = 6 4. Universal Set the set of ALL elements under consideration for a particular context. Also called a sample space 5. Subset a set whose elements all belong to another set the set notation symbol for subset is ex. Odd whole numbers is a sub-set of whole numbers: O W 6. Complement all elements of a universal set that do not belong to a given subset the set notation symbol for a complement is the set name and the prime symbol ‘. In general if A represents a sub-set then A’ represents the complement the sum of the number of elements in a set and its complement equals the universal set 𝑛(𝐴) + 𝑛(𝐴′ ) = 𝑛(𝑈) ex. If O is the set of odd whole numbers then O’ = {0, 2, 4, 6, 8, …} Page 1 of 2 Chapter 1: Set Notation 7. Math 3201 Handout #1 Empty Set a set with no elements the set notation symbol for empty set is { } ex. the set of odd numbers divisible by 2 is an empty set 8. Disjoint Sets two or more sets that have no elements in common ex. the set of even numbers and the set of odd numbers are disjoint sets “events” that describe disjoint sets are called “mutually exclusive” (these terms will be used the probability unit) 9. Venn Diagram a way of picturing relationships between different groups of things (sets/subsets) Named for the person who created it...John Venn universal set: rectangle ; represents everything in context of the problem sub-sets: circles inside the rectangle each element in the universal set occurs only once. if an element occurs in more than one subset then it is shown in overlapping subsets. These sections show elements that subsets have in common; called the “intersection” of two sets. (Section 1.2) We will work with 2-set and 3-set Venn diagrams in this course The diagram below to the left is a 2-set Venn diagram Two Set Venn Diagram Three Set Venn Diagram The diagram above to the right is a 3-set Venn diagram When a Venn diagram shows disjoint sets, the circles do not overlap Two Set Venn Diagram (Disjoint Sets) Page 2 of 2