Relation

Definition

A relation is a set of ordered pairs.

It describes which objects are connected.

Example:

Relation: lessThan
Pairs: {(1, 2), (1, 3), (2, 3)}

The pair:

(1, 2)

means:

1 is less than 2

Relation = structured connection between objects.

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Relation As Selected Pairs

A Cartesian product gives all possible ordered pairs.

Example:

A = {1, 2}
B = {a, b}
A x B = {(1, a), (1, b), (2, a), (2, b)}

A relation selects some of those pairs.

Example:

R = {(1, a), (2, b)}

A relation is a subset of a Cartesian product.

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Relation vs Predicate

A predicate can describe a relationship.

Example:

GreaterThan(x, y)

A relation stores the pairs where the predicate is true.

Example:

GreaterThan = {(5, 3), (4, 2)}

Predicate gives the test; relation gives the true pairs.

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Relations Can Connect A Set To Itself

Some relations connect elements from one set to elements in the same set.

Example:

A = {1, 2, 3}
R = {(1, 2), (2, 3)}

Here R relates elements of A to other elements of A.

This kind of relation is common when studying ordering, reachability, and graph structure.

Relations can describe internal structure inside a set.

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Final Summary

• A relation is a set of ordered pairs.
• A relation describes which objects are connected.
• A relation is a subset of a Cartesian product.
• Predicates and relations are closely connected.
• Relations can connect elements across sets or within the same set.