Here is a simple example and I'm pretty sure that I formed it correctly:
$$(A \wedge A) \vee (\bar{A} \wedge B) = (A \vee \bar{A}) \wedge (A \vee B) = 1 \wedge (A \vee B) = (A \vee B)$$
But what about this one:
$$(A \wedge \bar{B} \wedge \bar{C}) \vee (\bar{A} \wedge B \wedge \bar{C}) \vee (\bar{A} \wedge \bar{B} \wedge C) \vee (A \wedge B \wedge C)$$
I have no idea where to start but I started by sort the variables:
$$(A \vee \bar{A} \vee \bar{A} \vee A) \wedge (\bar{B} \vee B \vee \bar{B} \vee B) \wedge (\bar{C} \vee \bar{C} \vee C \vee C)=$$
$$(A \vee \bar{A}) \wedge (\bar{B} \vee B) \wedge (\bar{C} \vee C) = 1 \wedge 1 \wedge 1 = 1$$
I can see it's totally wrong, I know it.. But how to do that correctly?
What I'm trying to do here actually is minimizing the complete term.
Here I made table but how can you read the shorter function from it and use XOR?
What makes you think you can sort the letters?
The formula you are looking at is true if all three variables are true, or if precisely one variable is true. Thus, if you use $\otimes$ to describe the "exclusive or" operation, your formula is equivalent to $$ A \otimes B \otimes C. $$ If you are limited to $\lor, \land, \neg$ I do not think there is a much shorter way to write the formula than what you wrote.