How do I construct a boolean circuit from a boolean function when I cancelled our one of the variables?

Question

I was given a truth table of four variables, let's call them A,B,C,D. I found every instance where the truth table was true and turned it into a function:

F = (-A^-B^-C^-D) v (-A^-B^-C^ D) v (-A^-B^ C^-D) v ( A^-B^-C^-D) v ( A^-B^-C^ D) v ( A^-B^ C^-D)

I simplified this function to this:

(B ^ -C) v (B ^ -D)

This was part a) of the question.

The second part of the questions asks to us to use the function we simplified and construct a boolean logic circuit, using only AND, OR, and NOT gates (using as few as possible). I know how to construct logic circuits, but I am a bit confused about one thing: how can I create a logic circuit using the variables when one of the variables (A) was cancelled out in the simplification. What happens to it? Is it just gone? I hope I have articulated this problem the best way I can, and that I can find a good answer for this soon.

Answer 1

The expression (B ^ -C) v (B ^ -D) is already equivalent to an AND/OR/NOT circuit:

OR(AND(B, NOT(C)), AND(B, NOT(D)))

This can be rewritten by going through the following steps:

(-B)^(-C v -D)
(-B)^-(C ^ D)
-(B v (C ^ D))

Written as circuit:

NOT(OR(B, AND(C, D)))

• Scorch: So what happens to the A component? Is it just not included in the circuit?
• Axel Kemper: A is not relevant and thus is omitted.
• Scorch: Okay, thank for clearing my confusion Axel