Steps to simplify a Boolean Expression

Question

Simplify: $(x \land y) \lor (x \land \neg y) \lor (\neg x \land y)$

I need to simplify this using the using properties going step by step. I keep ending up with $(x \land y)$ as the answer but when I map is out I get that is should be $(x \lor y)$. Any help would be appreciated I would like to know what I am doing wrong.

Answer 1

$(x \land y) \lor (x \land \lnot y) = x$, so $(x \land y) \lor (x \land \lnot y) \lor (\lnot x \land y) = x \lor (\lnot x \land y)$

$x = x \lor (x \land y)$, so $x \lor (\lnot x \land y) = x \lor (x \land y) \lor (\lnot x \land y) $

$( x \land y ) \lor (\lnot x \land y) = y$, so $x \lor ( x \land y ) \lor (\lnot x \land y) = x \lor y$

Answer 2

First, we'll simplify the first two terms to $x$, then we'll simplify the entire statment $x \lor (\lnot x \land y)$:

$$(x \land y) \lor (x \land \lnot y) \equiv x \land (y \lor \lnot y) = x$$

$$x \lor (\lnot x \land y)\equiv (x \lor \lnot x) \land (x \lor y) \equiv x \lor y$$