Subtraction

[Ashley]

Hey,

I am doing the ALU part. Everything is fine but I don't understand only one single part:

Why X - Y equals ~(~X + Y)?

I understood that X - Y equals X + (-Y), where (-Y) is represented as two's compliment, which is toggle all the bits in Y and then plus 1.

But I can't see any equality in adding (-Y)'s two's compliment and ~(~X + Y).

Please give me some hint,
Thanks,
Ashley

[Koen_Be3840]

                                          ~y               ~x

When x = 3 -> 0011 ( 3) -> 0011 ( 3) -> 1100 (-4)
and    y = 1 -> 0001 ( 1) -> 1110 (-2) -> 0001 ( 1)
----------------------------------------------------------
                                        0001 ( 1) -> 1101 (-3) -> 0010 ( 2)

                                                                              ~(~x+y)

View also negation

Koen

[cadet1620]

Ashley wrote

I understood that X - Y equals X + (-Y), where (-Y) is represented as two's compliment, which is toggle all the bits in Y and then plus 1.

But I can't see any equality in adding (-Y)'s two's compliment and ~(~X + Y).

Here's an algebraic proof.

-a	=	~a+1
-a-1	=	~a
x-y	=	~(~x+y)
	=	-(-x-1+y)-1
	=	x+1-y-1
x-y	=	x-y

--Mark