How to simplify $\bar{X_1}\cup(\bar{X_2}\setminus(X_3\cap\bar{X_1})$?

Question

$$\bar{X_1}\cup(\bar{X_2}\setminus(X_3\cap\bar{X_1})$$

I was thinking of

But I am not sure if it will work for this one

Answer 1

Sure, that helps, since then you get:

$$\bar{X_1}\cup(\bar{X_2}\setminus(X_3\cap\bar{X_1})=\bar{X_1}\cup(\bar{X_2}\setminus X_3)\cup (\bar{X_2}\setminus \bar{X_1})$$

But we also have that:

$$A\cup (B\setminus A)=A\cup B$$

and:

$$A\cup (A\setminus B)=A$$

and so:

$$\bar{X_1}\cup(\bar{X_2}\setminus(X_3\cap\bar{X_1})=\bar{X_1}\cup(\bar{X_2}\setminus X_3)\cup (\bar{X_2}\setminus \bar{X_1})=\bar{X_1}\cup(\bar{X_2}\setminus X_3)\cup \bar{X_2}=\bar{X_1}\cup \bar{X_2}$$