Set Difference as Intersection with Complement

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Theorem

Set difference can be expressed as the intersection with the set complement:

$A \setminus B = A \cap \complement \left({B}\right)$


Proof

This follows directly from Set Difference as Intersection with Relative Complement:

$A \setminus B = A \cap \complement_S \left({B}\right)$.

Let $S = \Bbb U$.

Since $A, B \in \Bbb U$ by the definition of the universe, the result follows.

$\blacksquare$


Sources