Boolean Algebra: Convert F(a,b,c,d) = a’bd + ac’d

boolean-algebralogic

I'm attempting to simply convert the following boolean expression to using only NOR logic:
f(a,b,c,d) = a'bd + ac'd

The correct answer is
NOR( NOR(a',c'), NOR(a,b), d')

I've converted the solution to the following to compare to my work:
NOR( ac, a'b', d')

The following are two attempts which lead to the same scenario of being stuck:
Image of My Work

If I can get rid of the cb' term then I believe I'll have the right answer, but I'm unable to figure out how. Any help would be appreciated.
*This is a practice exam not hw

Best Answer

The Consensus Theorem states that:

$xy+x'z+yz=xy+x'z$

So, applied to your expression:

$ac+a'b'+b'c=ac+a'b'$

So yes, you can indeed get rid of the $b'c$ term.

If you don't like to use the consensus Theorem directly:

$ac+a'b'+b'c=ac+a'b'+b'c1=ac+a'b'+b'c(a+a')=$

$ac+a'b'+ab'c+a'b'c=ac(1+b')+a'b'(1+c)=ac1+a'b'1=ac+a'b'$

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