Sets

Entry conditions

Use set-based structures only when you can:

• Identify a finite or well-defined collection of elements.
• Decide membership clearly (no ambiguous membership rules).
• Represent relations as explicit pairs of elements.

If membership is undefined or dynamic without rules, do not treat it as a set.

Definitions

• A set $S$ is a collection of elements with clear membership.

Vocabulary (plain language)

• Set: a named collection of distinct items.
• Element: one item that may belong to a set.
• Ordered pair: a two-item list where order matters, written $(x,y)$.
• Membership: the rule that decides whether an element is in the set.

Symbols used

• $S, T$: sets.
• $x \in S$: “$x$ is an element of $S$.”

Intuition

Think of a set as a named bucket of distinct items. What matters is whether an item is in the bucket or not.

If you cannot list or decide membership, you should not claim a set. If you cannot list or decide the pairs, you should not claim a relation. If one input can map to multiple outputs, you do not have a function.

Worked examples

Example 1: A set

Let $S = \{\text{red}, \text{green}, \text{blue}\}$. Membership is clear: “red in $S$” is true, “yellow in $S$” is false.

Example 2: A set of strings

Let $S=\{\text{"cat"}, \text{"dog"}\}$. Then “cat” is in $S$, and “bird” is not.

How to recognize the structure

• Set: you can answer “is $x$ in $S$?” for any candidate $x$.

Common mistakes

• Treating vague groupings as sets when membership is not defined.

Minimal data

• A list of elements for $S$.

Misuse warnings

• Do not treat “similarity” as a function unless it is single-valued.
• Do not infer relations from vague description without listing pairs.