Extended Transitivity

Theorem

Let $S$ be a set.

Let $\RR$ be a transitive relation on $S$.

Let $\RR^=$ be the reflexive closure of $\RR$.

Let $a, b, c \in S$.

Then:

\(\text {(1)}: \quad\)

\(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c}\)

\(\implies\)

\(\ds a \mathrel \RR c\)

\(\text {(2)}: \quad\)

\(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel {\RR^=} c}\)

\(\implies\)

\(\ds a \mathrel \RR c\)

\(\text {(3)}: \quad\)

\(\ds \paren {a \mathrel {\RR^=} b} \land \paren {b \mathrel \RR c}\)

\(\implies\)

\(\ds a \mathrel \RR c\)

\(\text {(4)}: \quad\)

\(\ds \paren {a \mathrel {\RR^=} b} \land \paren {b \mathrel {\RR^=} c}\)

\(\implies\)

\(\ds a \mathrel {\RR^=} c\)

Proof

$(1)$ follows from the definition of a transitive relation.

$(4)$ follows from Reflexive Closure of Transitive Relation is Transitive.

Suppose that:

$\paren {a \mathrel \RR b} \land \paren {b \mathrel {\RR^=} c}$

By the definition of reflexive closure:

$b \mathrel \RR c$ or $b = c$

If $b = c$, then since $a \mathrel \RR b$:

$a \mathrel \RR c$

If $b \mathrel \RR c$ then by transitivity of $\RR$:

$a \mathrel \RR c$

Thus $(2)$ holds.

A similar argument proves that $(3)$ holds as well:

Suppose that:

$\paren {a \mathrel {\RR^=} b} \land \paren {b \mathrel \RR c}$

By the definition of reflexive closure:

$a \mathrel \RR b$ or $a = b$

If $a = b$, then since $b \mathrel \RR c$:

$a \mathrel \RR c$

If $a \mathrel \RR b$ then by transitivity of $\RR$:

$a \mathrel \RR c$

Thus $(3)$ holds.

$\blacksquare$