Convention

A set is not its element, and ${ϕ}$ is not $ϕ$

Algebra • Sets • Types of Sets

Statement

For any object

a

, the singleton

{a}

is a set with exactly one member and is not the same thing as

a

; the empty set

ϕ

has no members, while

{ϕ}

has one, so

ϕ \neq = {ϕ}

, and a set such as

{0}

is not empty because

0

is an element, not an absence of one.

How to use it

1Solve the defining condition first, then write the solution in braces before comparing it with anything.
2Count members: $ϕ$ has none, while ${ϕ}$ and ${0}$ each have one.
3Write $\in$ between an element and a set and $=$ or $\subset$ between two sets; an equality with a bare number on one side is a category error.

Write

{x : 3 x = 0}

in roster form, and say whether it equals

ϕ

Step-by-step

1
$3 x = 0 \Rightarrow x = 0$ , so the set is ${0}$
Why: The condition has exactly one solution.
2
${0}$ has one member and $ϕ$ has none, so ${0} \neq = ϕ$

Answer

{0}

, which is not

ϕ

Watch out

A set whose only member is

0

is not empty — that is the commonest error in this family — and neither is a set whose only member is the empty set.

Also known as

singleton versus its element, empty set versus set containing the empty set, {0} is not the null set, roster form of a solution set