I am having some trouble creating a circuit to satisfy this truth table. The assignment is to create a two-level OR-AND and a two-level AND-OR circuit to satisfy the following truth table.

x=x(a,b,c)
a b c x
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1

The AND-OR circut was easy, and I got the following.

(ac) OR (a'b'c')

The OR-AND circuit is a bit harder, as I can't determine which OR circuits I can make that i can combine with an AND gate to produce the table. I've tried the gates (a+c)... (b XOR c')... and (a+b+c)', to give me:

a+c = X
b XOR c' = Y
(a+b+c)' = Z

a b c X Y Z
0 0 0 0 1 1
0 0 1 1 0 0
0 1 0 0 0 0
0 1 1 1 1 0
1 0 0 1 1 0
1 0 1 1 0 0
1 1 0 1 0 0
1 1 1 1 1 0

As you can see, I can't figure out a way to combine any of these with an AND gate to make the above truth table.

Can someone please let me know if there is a better way to approach this? Any help you can give would be greatly appreciated.

Thanks

 

I think I figured it out.

I get (b'+c)(a XOR c').