Combining of logical conditions in IF statement

Question

I just notice the if condition below:

//1
if ((i >= 0) != (j >= 0))
 return false;

is just a short way for:

//2
if((i>=0 && j < 0) || (i < 0 && j >=0))
 return false;

Going from 1. to 2. takes some time to figure out, but how do we deduce the logic to go from 2. to 1.?

Answer 1

If we call (i >= 0) something arbitrary like A and (j >= 0) something else like B then it stands to reason that (i < 0) is just Not A (often shown as !A) and (j < 0) would be Not B or !B

Meaning this block:

if((i>=0 && j < 0) || (i < 0 && j >=0))
 return false;

Can be represented as:

if ((A && !B) || (!A && B))
    return false;
else // you haven't specified this part but it's implied
    return true;

Now, if we approach this as:

How do we get to the true value?

Then you realise it's the same as:

if ((A && B) || (!A && !B))
    return true;
else 
    return false;

Then we can call (A && B) something else like C; so it becomes:

if (C || !C)
    return true;
else 
    return false;

So, expanding out again, we can get:

if (A && B)
    return true;
else if (!(A && B)) 
    return true;
else 
    return false;

So, that's:

if ((i >= 0) && (j >= 0))
    return true;
else if (!( (i >= 0) && (j >= 0) ) )
    return true;
else 
    return false; 

Which can evaluate to:

if (True && True)
    return true;
else if (False && False)
    return true;
else 
    return false;

Showing that it's as simple as:

if ( (i >= 0) == (j  >= 0) )
    return true;
else // if ( (i >= 0) != (j  >= 0) )
    return false;

So, we've gone around the houses but we've worked our way down from statement 2 to statement 1.

NOTE: Adam's answer is more concise and is accurate from a pure logic perspective; this is more to provide a broader overview as I know that some people learn better when they can see the process.