  We discuss the relative K-theory for a $C^{*}$-algebra, $A$, together with a
$C^{*}$-subalgebra, $A' \subseteq A$. The relative group is denoted
$K_{i}(A';A), i = 0, 1$, and is due to Karoubi. We present a situation of two
pairs $A' \subseteq A$ and $B' \subseteq B$ are related so that there is a
natural isomorphism between their respective relative K-theories. We also
discuss applications to the case where $A$ and $B$ are $C^{*}$-algebras of a
pair of locally compact, Hausdorff topological groupoids, with Haar systems.
