Engineering Simplest Logic Circuit for z=0 with Inputs a, b, c

[STEMucator]

Homework Statement

Draw the simplest logic gate or circuit for the description below. The inputs are $a, b, c$ and the output is $z$.

$z = 0$ only if $abc = 010$ or $abc = 110$

Homework Equations

The Attempt at a Solution

So i drew up a quick table, AND and OR'ed the required terms together and came up with this:

Is this okay? The input $a$ seems to have disappeared, which makes no sense to me. Unless its some sort of trick question where I only need 2 inputs since the output doesn't rely on $a$ at all.

 

[berkeman]

Have you tried checking your answer with a Karnaugh map? I get a different answer...

 

[berkeman]

Also, have you tried plugging the two values back into your answer...?

 

[STEMucator]

I'm trying to do this without a K-Map.

Plugging the values of $b$ and $c$ into my answer does provide the correct output, but I am unsure about the input $a$.

I don't believe the answer is correct, and I am unsure how to algebraically simplify this if my above attempt is incorrect.

 

[berkeman]

My apolgies! I missed in your OP that z=0 for those two terms. Your answer does look correct. Sorry for any confusion I caused...

 

[STEMucator]

My apolgies! I missed in your OP that z=0 for those two terms. Your answer does look correct. Sorry for any confusion I caused...

Thank you for the clear up. I was just slightly confused about the inputs, but the logic worked out that way I guess.

 

[NascentOxygen]

z= 0 only if abc=010abc = 010 or abc=110

You did well. An alternative is to write $\bar z\ =\ \bar a b \bar c\ +\ ab\bar c$

Then simplify this as you did with your expression earlier, to get z = ...

The answers will be the same. :)