Transitive Relation

Definition

A relation is transitive when connection can be carried through an intermediate element.

Basic form:

if a relates to b
and b relates to c
then a relates to c

As ordered pairs:

(a, b)
(b, c)
therefore (a, c)

Transitive relation = connection carries forward.

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Simple Example

The relation:

less than

is transitive.

If:

1 < 2
2 < 3

then:

1 < 3

The relationship carries through the middle value.

Transitivity lets chains imply direct connections.

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Non-Example

The relation:

is parent of

is not transitive.

If:

Ada is parent of Ben
Ben is parent of Cleo

it does not follow that:

Ada is parent of Cleo

Ada is a grandparent, not a parent.

Not every chain preserves the same relation.

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Transitivity In Ordered Pairs

Example relation:

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

Because (1, 2) and (2, 3) are present, transitivity requires (1, 3).

It is included.

Non-transitive example:

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

Here (1, 3) is missing.

Transitivity checks whether required shortcut pairs exist.

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Why Transitivity Matters

Transitivity supports reasoning over chains.

Example:

A depends on B
B depends on C

If dependency is treated transitively, then:

A depends on C

This kind of structure appears in ordering, hierarchy, reachability, and computation graphs.

Transitivity turns step-by-step links into longer structural consequences.

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Final Summary

• A transitive relation carries connection through an intermediate element.
• If (a, b) and (b, c) are present, (a, c) must be present.
• "Less than" is transitive.
• "Is parent of" is not transitive.
• Transitivity supports reasoning over chains.
• Transitivity is important for order, dependency, and graph structure.