Conversion into disjunctive normal form

Question

Could someone possibly help me with a conversion into DNF. This is what I have done so far:

$$ x_{1} \wedge (x_2 \wedge(x_1 \vee x_3) \vee (x_3 \wedge \neg x_2)) \\ x_1 \wedge ((x_2 \wedge x_1) \vee (x_2 \wedge x_3) \vee (x_3 \wedge \neg x_2))$$

Then I am not sure where to go from here.

Through wolfram alpha I can see that the final result however is: $$(x_1 \wedge x_2) \vee (x_1 \wedge x_3)$$

Thanks in advance.

Answer 1

Here is the answer. Notice that $\land$ has bigger precedence than $\lor$, so sufficient parentheses are removed.

$$\begin{align} x_1\land (x_2\land (x_1\lor x_3)\lor x_3\land \neg x_2)&=x_1\land (x_1\land x_2\lor x_2\land x_3\lor x_3\land \neg x_2)\\&=x_1\land (x_1\land x_2\lor x_3\land (x_2\lor \neg x_2))\\&=x_1\land (x_1\land x_2\lor x_3)\\&=x_1\land x_1\land x_2\lor x_1\land x_3\\&=x_1\land x_2\lor x_1\land x_3 \end{align}$$