Efficient Boolean Minimization Techniques: Simplifying Y = \bar{X}_1+\bar{X}_0

[CentreShifter]

This is just a general question regarding Boolean minimization.

Expression:
$$Y=\bar{X}_1\bar{X}_0+\bar{X}_1X_0+X_1\bar{X}_0$$

Minimized expression:
$$Y=\bar{X}_1+\bar{X}_0$$

My first attempt was to minimize it algebraically. I factored $$\bar{X}_1$$ from the first two terms, then the $$\bar{X}_0+X_0$$ reduce to 1. So I end up with $$\bar{X}_1+X_1\bar{X}_0$$.

My question then is, how does the second term (last term from the original expression) reduce to $$\bar{X}_0$$ to end up with the known correct expression?

 

[vela]

Multiply the first term by

$$\bar{X}_0+1$$

 

[zgozvrm]

Try factoring out $$\bar{X_1}$$ from the first two terms using the distributive property.

Also, remember that in boolean algebra not only does multiplication (AND) distribute over addition (OR), but addition also distributes over multiplication. That is,

$$A(B + C) = AB + AC$$

and

$$A + (B C) = (A + B)(A + C)$$