# Set Builder Notation for Natural Numbers

by Misha
(U.S.A.)

Set Notation

The set shown below is the arithmetic sequence for all natural numbers.

Convert the set notation from Roster Notation to Set Builder Notation

{1, 2, 3, 4, 5, ...........}

### Comments for Set Builder Notation for Natural Numbers

 Aug 18, 2013 Convert Set Notation by: Staff Answer Part I Set “S” = the set of natural numbers greater than or equal to 1. (Natural numbers are whole counting numbers. Natural numbers do not include decimals or fractions.) Four different ways of representing the set S of all natural numbers: 1. ROSTER Notation 2. SET BUILDER Notation 3. INTERVAL Notation 4. GRAPHICAL Representation 1. The ROSTER Notation is given in the problem statement: S = {1, 2, 3, 4, 5, ...} The first five elements (1, 2, 3, 4, and 5) establish the pattern. The ellipsis (three dots) shows that the pattern continues indefinitely. --------------------------------------------

 Aug 18, 2013 Convert Set Notation by: Staff -------------------------------------------- Part II 2. SET BUILDER Notation: There is more than one format for writing this sequence in Set Builder Notation. Here is a commonly used format: S = {n | n ∈ ℕ, n ≥ 1} {} curly brackets surround the expression ∈ = element of a set | and : can be used interchangeably. Both notations are separators which mean “where” or “such that” n: the first “n” is the “output function”, shown as = {n | n: the second “n” is the “variable”, shown as | n ∈ N, ℕ (the set of all natural numbers) is the“input set” n ≥ 1 is the “predicate” Reading from left to right: “S” is the set of all numbers “n” {n | … , …} where “n” is an element of the set of natural numbers ℕ { … | n ∈ ℕ, … } and “n” is greater than or equal to 1 {… | … , n ≥ 1}. 3. INTERVAL Notation: S = [1 . . ∞) This is an “unbounded interval”. The endpoint of 1 is included in the interval, but the endpoint ∞ is not included in the interval. Therefore [ ) must be used to show the interval. However, this notation CANNOT BE USED: S = [1, ∞). [1, ∞) means all REAL numbers which are greater than or equal to 1. Your set includes only INTEGERS. Your set does not include any other real numbers. Interval Notation for Integers should be written: S = [1 . . ∞) --------------------------------------------

 Aug 18, 2013 Convert Set Notation by: Staff -------------------------------------------- Part III It should be read: the set of all natural numbers beginning with (and including) 1. 4. GRAPHICAL Representation: S = {1, 2, 3, 4, 5, ...} Thanks for writing. Staff www.solving-math-problems.com

 Feb 08, 2016 Interval Notation should not be used for a discrete set NEW by: Anonymous The use of interval notation is very specific and would include all real numbers between the two numbers (and possibly the numbers themselves). The interval [1,infinity) includes numbers like 2/3, 5.287, and pi. None of these are natural numbers.