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Glacius
FoundationsConcept reference

Set membership

Set membership states whether a specified object is one of a set’s elements.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. A finite set is a collection whose members are specified
  3. The empty set, varnothing, contains nothing
  4. Check your understanding
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01A finite set is a collection whose members are specified#

A finite set is a collection whose members are specified. In A={2,5,8}A=\{2,5,8\}, the braces enclose the member labels.

Set A encloses the labels 2, 5, and 8. The label 4 is outside A. Enclosure, not position in a list or colour, determines membership.Set A encloses the labels 2, 5, and 8. The label 4 is outside A. Enclosure, not position in a list or colour, determines membership.
Figure 1The boundary separates members from a candidate outside.
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Let A={2,5,8}A=\{2,5,8\}. To test membership, look for the object in this roster. The label 55 is listed and 44 is absent.

Membership notation.

5∈A,4∉A5\in A,\qquad4\notin A

The symbol ∈\in means “belongs to”; ∉\notin means “does not belong to.” Order and repetition do not change which objects belong. {2,5,8}\{2,5,8\} and {8,2,5}\{8,2,5\} name the same set.

Check your reasoning

Let A={2,5,8}A=\{2,5,8\}. Among the candidate labels 3, 5, 7, enter the one that belongs to AA.

Show answer and explanation
5

The roster explicitly includes 5. Neither 3 nor 7 is listed.

02The empty set, varnothing, contains nothing#

The empty set, ∅\varnothing, contains nothing. The set {0}\{0\} contains one object: the label 00. Zero is not a special exception to membership rules.

These membership claims differ.

0∉∅,0∈{0}0\notin\varnothing,\qquad0\in\{0\}
Check your reasoning

An allowed-format set is {csv, json}. Sort these labels.

Show answer and explanation

Only csv and json are listed. Similarity to another file label does not create membership.

03Check your understanding#

Check your reasoning

Labels 4, 4, 9 were observed. Which set records them?

  1. A{4,9}\{4,9\}
  2. B{4}\{4\}
  3. C{9}\{9\}
Show answer and explanation
{4,9}\{4,9\}

Repeating 4 adds no new member: keep 4 and 9.

Key takeaway

Check the specified members. Order and repeated mentions do not change membership, and the empty set contains no object, including zero.

  • Decide whether an object belongs to a specified finite set

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Set membership.” Math behind ML. /learn/f-sets