# Thread: Set theory proof A-(B-C)=(A-B)-(A-C)

1. ## Set theory proof A-(B-C)=(A-B)-(A-C)

I just took a quiz in math and the problem asked me to prove A-(B-C)=(A-B)-(A-C). I couldn't figure it out and don't want to wait until after spring break to find out what I was supposed to do. My professor said I was overthinking it and I couldn't even get started. Please help

2. ## Re: Set theory proof A-(B-C)=(A-B)-(A-C)

This isn't generally true. An element of A that isn't in either B or C is on the left but not the right.

3. ## Re: Set theory proof A-(B-C)=(A-B)-(A-C)

Originally Posted by JT3063
I just took a quiz in math and the problem asked me to prove A-(B-C)=(A-B)-(A-C). I want to wait until after spring break to find out
I use A\setminus B, $A\setminus B$ for set difference,
The bad news is it not true.
EX> $U=\{a,b,c,d,e,f\}$, $A=\{a,b,e\},~B\{b,d,f\},~\&~C=\{a,c,f\}$
\begin{align*}A\setminus B&=\{a,e\} \\A\setminus C&=\{b,e\}\\B\setminus C&=\{b,d\} \\A\setminus(B\setminus C)&=\{a,e\}\\\text{BUT }( A\setminus B)\setminus (A\setminus C) &=\{a\} \end{align*}

4. ## Re: Set theory proof A-(B-C)=(A-B)-(A-C)

One thing that is true is
$A \cap (B - C) = (A \cap B) - (A \cap C)$