How does this Boolean simplification work?

Question

The following expression:

$$ABC + BCD + A'D$$

simplifies to:

$$ABC + A'D$$

I just cannot understand how this comes about.

Answer 1

Treat $BC=X$ as a single variable. Then: $$ AX + XD + A'D = AX + (A+A')XD +A'D = AX + AXD + A'XD + A'D \\ = AX(1+D) + A'D(1+X) = AX + A'D $$

Please tell if you did not understand anything.

Answer 2

Using a Karnaugh Map will make this a lot easier to understand. If you input your original statement into a 4x4 map, you end up with the following expression:

$$A'C'D + A'CD + ABC$$

simplifies to:

$$A'D + ABC$$

Here is the Wikipedia article on Karnaugh Maps: https://en.wikipedia.org/wiki/Karnaugh_map.

Edit: The Karnaugh Map actually outputs the fully-simplified expression ABC + A'D, I just wanted to show where in the process the BCD term disappeared.

Answer 3

$ABC + BCD + A'D = ABC + ABCD + A'BCD + A'D = ABC + A'D$