Simplify 4-term Boolean Algebra expression

Question

How do I get from this:

$F = AB' + AC' + AD + C'D'$

to this:

$F = AB' + AD + C'D'$

Not sure how the $AC'$ disappeared.

Answer 1

When $AC'$ is to be true, both $A$ and $C'$ must be true. Then either $AD$ is true or $C'D'$, depending on whether $D$ is true or not.
In other words $AC'$ implies $(AD + C′D′)$, so the first term gets 'assimilated' into the sum:

$AC′ + AD + C′D′ $

$= AC′1 + AD + C′D′$

$= AC′(D+D') + AD + C′D′$
$= AC′D + AC'D' + AD + C′D′$
$= A(C'+1)D + (A+1)C'D'$
$= A1D + 1C'D'$
$ = AD + C′D′$