Question

Prove a set Theory identity

Answer 1

\[A\cap(B-C)=(A\cap B) - (A \cap C)\]

Answer 2

To show equality, you need to show that \[A\cap(B-C)\subseteq(A\cap B) - (A \cap C)\]and that \[(A\cap B) - (A \cap C)\subseteq A\cap(B-C)\]

Answer 3

Typo there. One minute.

Answer 4

To do this, suppose \(x∈A∩(B−C)\). That means it's in A, and in \(B−C\). So \(x∈B\) and \(x∉C\)

Answer 5

ok

Answer 6

That means that \(x\in A\cap B\), and \(x\notin A\cap C\). It follows that \[x\in((A\cap B) - (A \cap C))\]so \[A\cap(B-C)\subseteq(A\cap B) - (A \cap C)\]

Answer 7

Now for the other way. Suppose \[x\in((A\cap B) - (A \cap C))\]Then \(x\in(A\cap B)\) and \(x\notin (A \cap C)\). The means that \(x\in A\), \(x\in B\)

Answer 8

The easiest approach is to use Venn diagram.

Answer 9

Also, we know that either \(x\notin A\) or \(x\notin C\). However, \(x\in A\), so it must be \(x\notin C\). Hence, since \(a\in B\) and \(x\notin C\) we have that \(x\in(B-C)\). Since it's in \(A\), we have that \(x\in A\cap(B-C)\). Therefore, \[(A\cap B) - (A \cap C)\subseteq A\cap(B-C)\]

Answer 10

Since both sides are subsets of each other, we have equality.

Answer 11

YAY thanks

Answer 12

That was awesome

Answer 13

You're welcome.

Answer 14

I just posted this question on a site that i subscribed and everyone messed up badly lol. I guess OS Rocks

Answer 15

Thanks

Answer 16

OS does rock :)

Although I am very prone to messing up. I did mess up on this one, but fortunately we both caught it.

Answer 17

Nah u r awesome

Answer 18

I'm honored :)