Set

Definition

A set is a collection of distinct objects.

Example:

{1, 2, 3}

This set contains the objects:

1
2
3

Set = collection of distinct elements.

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Sets Group Objects

Sets let us treat many objects as one structured collection.

Example:

Even numbers under 10 = {2, 4, 6, 8}

Instead of listing each object separately every time, we can refer to the whole collection.

A set lets us reason about many objects together.

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Order Does Not Matter

In a set, order does not matter.

These describe the same set:

{1, 2, 3}
{3, 2, 1}

The elements are the same.

Only the written order changed.

Sets care about membership, not order.

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Repetition Does Not Matter

Repeating an element does not create a new element in the set.

Example:

{1, 1, 2, 3}

describes the same set as:

{1, 2, 3}

The element 1 either belongs to the set or it does not.

Sets contain distinct elements.

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Sets As Domains

Sets can define the domain of a logical statement.

Example:

For every x in {1, 2, 3}, x > 0.

The set tells us which values x can take.

Sets make domains explicit.

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Final Summary

• A set is a collection of distinct elements.
• Sets group objects into one structure.
• Order does not matter in a set.
• Repetition does not matter in a set.
• Sets can define domains for variables and predicates.
• Sets are the foundation for relations and functions.