# Set Difference of Intersection with Set is Empty Set

## Theorem

The set difference of the intersection of two sets with one of those sets is the empty set.

Let $S, T$ be sets.

Then:

$\left({S \cap T}\right) \setminus S = \varnothing$
$\left({S \cap T}\right) \setminus T = \varnothing$

## Proof

$\left({R \cap S}\right) \setminus T = \left({R \setminus T}\right) \cap \left({S \setminus T}\right)$

Hence:

 $\displaystyle \left({S \cap T}\right) \setminus S$ $=$ $\displaystyle \left({S \setminus S}\right) \cap \left({T \setminus S}\right)$ Set Difference is Right Distributive over Set Intersection $\displaystyle$ $=$ $\displaystyle \varnothing \cap \left({T \setminus S}\right)$ Set Difference with Self is Empty Set $\displaystyle$ $=$ $\displaystyle \varnothing$ Intersection with Empty Set

$\Box$

 $\displaystyle \left({S \cap T}\right) \setminus T$ $=$ $\displaystyle \left({T \cap S}\right) \setminus T$ Intersection is Commutative $\displaystyle$ $=$ $\displaystyle \varnothing$ from above

$\blacksquare$