I need prove a boolean function

Question

In need to prove with boolean algebra that XOR complement (negado) is equal to XNOR but i cant do it, can you help me?

!(!xy+x!y)=xy+!x!y

how to prove it?

Answer 1

If + means or and ! means not, and juxtapose means and (like ab for a and b) then DeMorgans rules become !(a+b)=(!a)(!b) and $!(ab)=(!a)+(!b).$

Then $!(!xy +x!y)=(!(!x)y)(!(x!y))$ Doing DeMorgans inside both (and using $!!a=a$) then makes it $(x+!y)(!x+y).$ Distributing then gives four terms, two of which ($x!x$ and $!yy$) are zero and only $xy+!x!y$ remains.