The top left in my k-map is a 0.

(Question is: Design a 4-Bit Odd Parity Checker, output is 1 when exactly odd number of inputs (A,B,C,D is on )

I have got down to

\(

A\cdot \overline B \cdot \overline C \cdot \overline D + \overline A \cdot \overline B \cdot \overline C \cdot D + A \cdot B \cdot \overline C \cdot D + \overline A \cdot B \cdot C \cdot D+ A \cdot \overline B \cdot C \cdot D + \overline A \cdot \overline B \cdot C \cdot \overline D + A \cdot B \cdot C \cdot \overline D

\)

I am not comfortable simplifying this kind of thing at all, this is my first year at Uni after 5 years away from school! I think the fact that I don't remember "factoring" in decimal maths is what's holding me back here.

I've been able to solve/simplify much simpler expressions, but this has just stumped me, I don't know where to start and feel like every time I make an attempt I'm just "guessing" my way through it, and going the wrong wrong way about it!

I just need someone to give me a little push in the right direction, because I know the beginning and I have an idea of the end but I can't get through the middle...

Also, XOR and XNOR are the two functions that we just barely touched but that I have not been able to get my head around. I understand the whole "equality gate"/"inequality gate" idea but I do not understand how to recognize this in an expression or put the function to use. I haven't been able to find anything detailed online, either.

I was simply told if you see

\(

\overline A \cdot B + A \cdot \overline B

\)

then you "know" its an XOR function. How do I "prove" this? What about XNOR?

Sorry about the big post, I hope someone can point me in the right direction! Thanks for readin if you've gotten the whole way down here