An Isomorphism That Does Not Give Rise to a Congruence Relation
\[V=\mathbb{R}^3\]
and let\[S= \{(a,0,b):a,b \in \mathbb{R} \}\]
\[S= \{(0,0,b):a,b \in \mathbb{R} \}\]
Now
\[S, \: \subset V\]
and \[S\]
and \[T\]
are isomorphic.\[\mathbb{R} \cap T = \emptyset\]
Hence
\[\mathbb{R}^3\]
can be decomposed into the direct sum of disjoint subspaces of \[\mathbb{R}^3\]
, \[\mathbb{R}^3 = \mathbb{R} \oplus T\]
.We can not write
\[S\]
in this way.Hence
\[S, \: T\]
are isomorphic but do not obey any congruence relation. 
