Element

Definition

An element is an individual object that belongs to a set.

Example:

Set: {1, 2, 3}
Element: 2

The object 2 is an element of the set because it is one of the objects listed inside it.

Element = one object inside a set.

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Element vs Set

An element is a single object.

A set is a collection of objects.

Example:

1

is an element.

{1, 2, 3}

is a set.

The set contains elements.

The element is not the same thing as the set that contains it.

Element and set are different levels of structure.

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Elements Can Be Different Kinds Of Objects

Elements do not have to be numbers.

Examples:

{cat, dog, bird}

Elements:

cat
dog
bird

Another example:

{true, false}

Elements:

true
false

A set can contain any clearly distinguishable objects.

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Why Elements Matter

Logic uses variables and predicates.

Sets give those variables a domain of possible elements.

Example:

Domain: {1, 2, 3}
Predicate: Even(x)

The variable x may refer to an element of the domain.

If:

x = 2

then x refers to one element of the set.

Elements are the objects that variables can range over.

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Final Summary

• An element is an individual object in a set.
• A set is a collection; an element is one member of that collection.
• Elements can be numbers, symbols, truth values, or other objects.
• Sets define domains of possible elements.
• Elements are the objects that predicates, relations, and functions operate on.