**Consider the variables x,y and z which are related as follows:**

*-> z = x (XOR) y*

Are

**'***x'***and '***z'*__dependent__*or*__independent__?

**( x and y can take the values [0,1]**

**XOR stands for the eXclusive-OR operator. )**

For all possible ordered pairs

*, we get a corresponding*

**(x,y)***.*

**z**

__x__

__y__

__z__( 0 , 0 ) -> 0

( 0 , 1 ) -> 1

( 1 , 0 ) -> 1

( 1 , 1 ) -> 0 ... (A)

We obtain a certain value of z according to the values of x and y.

So, it might seem intuitive that x and z, or y and z, are DEPENDENT, i.e., z changes when either x or y change.

Now, consider the ordered pairs

**.**

*(x,z)*We have,

(0,0) , (0,1) , (1,1) and (1,0) ... (from A)

These are

*all the possible ordered pairs*of (x,z) just like (x,y), and each of them occurs with equal probability.

This means that just like all ordered pairs (x,y), which were independent,

*all the ordered pairs (x,z) are also*

__independent__

**and not dependent, unlike what intuition suggests.**

*,*Note : The word 'independent' might be deceptive, as such, but what we are looking at here is the Mathematical meaning of it and its usage in proofs et al.