# Thread: Distributive property for the difference of sets

1. ## Distributive property for the difference of sets

Again thanks for all the help I've been getting on this wonderful site.

I could not find this property in my textbook, yet it has to be used to solve the following step:

[ ($\displaystyle A_1 \cap A_2 \cap ... \cap A_k$ - B) ] $\displaystyle \cap$ ($\displaystyle A_k$$\displaystyle _+$$\displaystyle _1$ - B)

Is there a distributive property for the difference of sets that is used to write the following?:

[ ($\displaystyle A_1 \cap A_2 \cap ... \cap A_k$ ) $\displaystyle \cap$ $\displaystyle A_k$$\displaystyle _+$$\displaystyle _1$] - B

2. Originally Posted by oldguynewstudent
[ ($\displaystyle A_1 \cap A_2 \cap ... \cap A_k$ - B) ] $\displaystyle \cap$ ($\displaystyle A_k$$\displaystyle _+$$\displaystyle _1$ - B)
Is there a distributive property for the difference of sets that is used to write the following?:
[ ($\displaystyle A_1 \cap A_2 \cap ... \cap A_k$ ) $\displaystyle \cap$ $\displaystyle A_k$$\displaystyle _+$$\displaystyle _1$] - B
Yes of course.
$\displaystyle (A\cap B \setminus E)\cap (C\setminus E)$
$\displaystyle =(A\cap B) \cap E^c)\cap (C\cap E^c)$
$\displaystyle =(A \cap B \cap C) \cap E^c$
$\displaystyle =(A \cap B \cap C) \setminus E$