Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was understanding this proof :

(A-B) X C = (A X B) - (B X C)

A transition in statements in proof seemed so wrong to me , which are following :

$(x \in A \;and\; x \notin B) \;and \; y \in C$

$(x \in A\; and \;y \in C) \;and\; (x \notin B\; and \;y \in C)$

$(x,y) \in (A X C) \;and\; (x,y) \notin (B X C)$

my question lies in last two statements , how can we write $(x\notin B\;and\;y \in C)$ to $(x,y) \notin (B X C)$ ?

I mean statement $(x \notin B \;and\; y \in C)$ says x doesn't belongs to B and y belongs to C.

It's equivalent (in proof) statement says (x,y) such that x doesn't belongs to B and y doesn't belongs to C , isn't that a contradiction or am I missing something here ?

share|cite|improve this question
up vote 2 down vote accepted

$B\times C$ is the set of all ordered pairs $\langle b,c\rangle$ such that $b\in B$ and $c\in C$. If $x\notin B$, the pair $\langle x,y\rangle$ cannot be in $B\times C$, because its first component, $x$, is not in $B$. it doesn’t matter whether $y\in C$ or $y\notin C$: the fact that $x\notin B$ is already enough to guarantee that $\langle x,y\rangle\notin B\times C$.

As an example, suppose that $B=\{1,2\}$ and $C=\{3,4,5\}$. Then $\langle 3,4\rangle\notin B\times C$, because $3\notin B$; $\langle 1,6\rangle\notin B\times C$, because $6\notin C$; $\langle 1,4\rangle\in B\times C$ because $1\in B$ and $4\in C$; and $\langle 3,6\rangle\notin B\times C$ both because $3\notin B$ and because $6\notin C$: it fails for two reasons to belong to $B\times C$.

share|cite|improve this answer
awesome explanation, Thanks a lot :) – Mr.Anubis Sep 17 '12 at 9:34

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.