A set is a collection of things where there is never any argument about whether something belongs to it. The things in a set are called its elements. The rule that decides membership must be so clear that any two people applying it would always agree.
The tall students in this class is not a set, because two teachers would draw the line in different places. The students taller than 150 cm is a set, because a tape measure settles every case.
How to write a set
| Symbol | Meaning | Example |
|---|---|---|
| { } | Curly brackets hold the elements | A = {2, 4, 6, 8} |
| Capital letter | Names the set | A, B, C |
| ∈ | Is an element of | 4 ∈ A |
| ∉ | Is not an element of | 5 ∉ A |
| n(A) | The cardinal number: how many elements | n(A) = 4 |
| { } or ∅ | The empty set, with no elements at all | n(∅) = 0 |
Order does not matter, so {1, 2, 3} and {3, 1, 2} are the same set. Repeats do not count, so {a, a, b} is just {a, b} and has two elements, not three.
Two ways to describe the same set
| Method | How it works | Example |
|---|---|---|
| Listing method | Write every element inside the brackets | A = {1, 3, 5, 7, 9} |
| Description method | State the rule in words instead of listing | A = the odd numbers under 10 |
Listing is clearer for a small set. Description is essential for a large or infinite one. Three dots are acceptable, as in {2, 4, 6, ..., 100}, provided the pattern is genuinely obvious from the first few terms.
Kinds of set
| Name | What it means |
|---|---|
| Finite set | The counting stops |
| Infinite set | The counting never stops |
| Empty set | No elements at all |
| Equal sets | Exactly the same elements |
| Equivalent sets | The same NUMBER of elements |
| Subset | Every element of one is also in the other |
The set {0} contains one thing, namely the number zero, so it has one element. The empty set contains nothing, so it has none. An empty box and a box containing a piece of paper with 0 written on it are not the same box.
{1, 2, 3} and {a, b, c} are equivalent, because both have three elements, but not equal, because they contain different things. Every pair of equal sets is equivalent, but the reverse fails constantly.
Counting the subsets
A subset is any selection from a set, including taking everything and taking nothing. A set with n elements has 2 to the power n subsets, because each element faces one independent decision: in or out.
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Write the set of factors of 12 by the listing method, and state n(A).
List all the subsets of B = {a, b, c} and check the count.
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A set is a well defined collection: there is never any argument about what belongs.
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