What a set is: listing, set-builder, finite, infinite and empty sets
Mathematics Prep 1 Term 1, Unit 1, Lesson 3
GoalDescribe a set by listing or by a rule, and say whether it is finite, infinite or empty.
New words
set · a well-defined collection of different thingselement · one of the things in a setfinite set · a set whose elements can all be countedempty set · the set with no elements, written ∅ or { }
Explanation
A set is a collection of things that is well defined: you can always tell whether something is in it or not. We write its elements inside braces { }, and we do not repeat an element or care about the order, so {2, 2, 3} is the same set as {3, 2}. There are two ways to write a set: listing the elements, A = {2, 4, 6, 8}, or giving a rule, A = {x : x is an even number between 1 and 9}. In this book N = {0, 1, 2, 3, …}. A set is finite if its elements can be counted, and infinite if they go on forever, such as the multiples of 5. The empty set ∅ has no elements. It is not {0}, because {0} has one element.
★Remember
Braces { }. Each element once. Order does not matter. ∅ has no elements.
Worked examples
A is the set of even numbers between 1 and 9. Listing: A = {2, 4, 6, 8}. The same set in any order is {8, 6, 4, 2}. A set with repeated elements such as {2, 2, 3} is just {2, 3}.
Rule to list. B = {x : x ∈ N, 2 < x < 7} means all numbers x in N that are bigger than 2 and smaller than 7. So B = {3, 4, 5, 6}.
Set
Elements
Kind
days of the week
7 days
finite
multiples of 5
5, 10, 15, …
infinite
N between 3 and 4
∅
empty
Kinds of sets. The days of the week are finite (7 elements). The multiples of 5, {5, 10, 15, …}, are infinite. The numbers of N between 3 and 4 do not exist, so that set is empty: ∅.
Class activity
Set hunt. With a grown-up, find five things on a table and write them as a set. Then write a rule for another set, such as “the coins in my purse”. Is it finite? Can it be empty?
Practice
1
Write the set of odd numbers between 2 and 10 by listing.
{3, 5, 7, 9}.
2
List the elements: C = {x : x ∈ N, x < 4}.
N starts at 0, so C = {0, 1, 2, 3}.
3
Is {1, 2, 3, 4, …} (all counting numbers) finite or infinite?
Infinite: the numbers never stop.
4
Is the set of even prime numbers bigger than 2 empty?
Yes, ∅. The only even prime number is 2.
5
Youssef says ∅ = {0}. What is wrong?
{0} has one element, the number 0. ∅ has no elements.
