Simplification of a logical equation

[momentum]

here is what my book has given for simplification.

I'm not clear with that red marked part in the given solution. How come they got two same items in that red box ?

However, my solution is different

 

[BvU]

You can write X'Y = X'Y + X'Y : the truth tables left and right are identical ...

 

[momentum]

X'Y = X'Y + X'Y : the truth tables left and right are identical ...

Is it a Law ? which Law is this ?

 

[NascentOxygen]

Is it a Law ? which Law is this ?

You can OR something with itself and the expression doesn't change its value. If it had the value T then T+T is still True, and if it had the value F then F+F is still False.

 

[NascentOxygen]

Occasionally, it aids simplification if you first introduce an extra term without changing anything. Just as in algebra you can replace B by 1× B, or by (B + 0) or by B×W/W, then in Boolean algebra you can replace A by A + A or by A•True.

Now, in your cited example I can see no reason for doing this where they have. I expected the author was going to perform some neat move, but he doesn't. He just causes a couple of extra unnecessary lines, before using the fact that A + A’ B is equivalent to A + B.

This is something you should commit to memory, it comes in very handy. Demonstrate it using a Truth Table before trying to prove it using rules of Boolean algebra.

Apply this to further simplify the terms in your solution that you have circled: x’ y’ + x’ y z