Composition of Relations Preserves Subsets

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Theorem

Let $A, B, S, T$ be relations as subsets of Cartesian products.

Let $A \subseteq B$ and $S \subset T$.

Then:

$A \circ S \subseteq B \circ T$

Proof

We have:

\(\ds \forall \tuple{x, y}: \, \)

\(\ds \tuple{x, y} \in A \circ S\)

\(\leadsto\)

\(\ds \exists z : \tuple{x, z} \in S, \tuple{z, y} \in A\)

Definition of Composite Relation

\(\ds \)

\(\leadsto\)

\(\ds \exists z : \tuple{x, z} \in T, \tuple{z, y} \in B\)

Definition of Subset

\(\ds \)

\(\leadsto\)

\(\ds \tuple{x, y} \in B \circ T\)

Definition of Composite Relation

By definition of subset:

$A \circ S \subseteq B \circ T$

$\blacksquare$