I need to find the answer of this kind of expression $a_1 + a_2 + a_3 + (a_1\oplus a_2) + (a_2\oplus a_3) + (a_1\oplus a_3) + (a_1\oplus a_2\oplus a_3) $.
After observing I found this pattern in above expression $ X + (X \oplus Y)$. But I don't know how to proceed further.
$p \oplus q = (p+q)\cdot(p'+q') = p'q + pq'$
You can simplify XOR operations using the above relation, and further simply them using the laws of boolean algebra.
$a_1 + (a_1\oplus a_2) = a_1 + (a_1'a_2 + a_1a_2')$
$= a_1 + a_1a_2' + a_1'a_2 = a_1 + a_1'a_2$
Similarly, you can simplify other terms.
After simplification, the expression would be,
$= a_1 + a_2 + a_3 + (a_1 \oplus a_2 \oplus a_3)$
$= a_1 + a_2 + a_3 + (a_1'a_2 + a_1a_2)'a_3 + (a_1'a_2 + a_1a_2)a_3'$
$= a_1 + a_2 + a_3 + (a_1'a_2 + a_1a_2)a_3'$
$= a_1 + a_2 + a_3 + a_1'a_2a_3c + a_1a_2a_3'$
The law of boolean algebra used here is ABSORPTION LAW
The simplified expression is $a_1 + a_2 + a_3$