what is the cardinality of equivalence classes of relation $ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $?

Question

given :$$T\subseteq \mathbb{N} $$

$$ R=\{<A,B>\in P(\mathbb{N} )|A\cap T=B\cap T\} $$ what is a equivalence relation what is the cardinality of equivalence classes of relation R ? how can I prove it? I think it is $P(T)$

Answer 1

The cardinality of the set of equivalence classes is $|\mathbb{N}/R|=|\mathcal{P}(T)|$, this can be easily seen as $$\mathbb{N}/R=\{[B]:B\subset T\}$$ where $[B]$ denotes the equivalence class containing $B$.

The cardinality of individual equivalence classes is $|[B]|=|\mathcal{P}(\mathbb{N}\setminus T)|$. Again this can be easily seen as the equivalence class of some $B\subset T$ is given by $$[B]=\{B\cup X:X\in\mathcal{P}(\mathbb{N}\setminus T)\}.$$