Equivalence Relation on Power Set induced by Intersection with Subset

Theorem
Let $A, T$ be sets such that $A \subseteq T$.

Let $S = \powerset T$ denote the power set of $T$.

Let $\alpha$ denote the relation defined on $S$ by:
 * $\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$

Then $\alpha$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
We have that for all $X \in S$:


 * $X \cap A = X \cap A$

That is:


 * $X \mathrel \alpha X$

Thus $\alpha$ is seen to be reflexive.

Symmetry
Thus $\alpha$ is seen to be symmetric.

Transitivity
Let $X \mathrel \alpha Y$ and $Y \mathrel \alpha Z$.

Then by defintion:

Hence:
 * $X \cap A = Z \cap A$

and so by definition of $\alpha$:
 * $X \mathrel \alpha Z$

Thus $\alpha$ is seen to be transitive.

$\alpha$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.