Image of Set Difference under Relation/Corollary 1

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Corollary to Image of Set Difference under Relation

Let $\RR \subseteq S \times T$ be a relation.

Let $A \subseteq B \subseteq S$.


Then:

$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} \subseteq \RR \sqbrk {\relcomp B A}$

where:

$\RR \sqbrk B$ denotes the image of $B$ under $\RR$
$\complement$ (in this context) denotes relative complement.


Proof

We have that $A \subseteq B$.

Then by definition of relative complement:

$\relcomp B A = B \setminus A$
$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk B \setminus \RR \sqbrk A$


Hence, when $A \subseteq B$:

$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} \subseteq \RR \sqbrk {\relcomp B A}$

means exactly the same thing as:

$\RR \sqbrk B \setminus \RR \sqbrk A \subseteq \RR \sqbrk {B \setminus A}$

$\blacksquare$