Boolean Logic using proofs

Question

ABC' + C = AB + C

I understand this using venn diagrams and intuition. However, I am not able to derive the proof for getting from one side to the other. It's probably very simple step that I keep missing. Please enlighten me.

Answer 1

$$AB \overline C + C$$ Identity Law: $X • 1 = X$ $$AB \overline C + 1 • C$$ Annulment Law: $X + 1 = 1$ $$AB \overline C + (AB + 1) C$$ Distributive Law: $X • (Y + Z) = X Y + X Z$ $$AB \overline C + ABC + C$$ Distributive Law: X Y + X Z = $X • (Y + Z)$ $$AB (\overline C + C) + C$$ Complement Law: $X + \overline X = 1$ $$AB + C$$ $$AB \overline C + C = AB + C$$

Answer 2

Either $C$ is true or false.

If $C$ is true:

$ABC' + C = C$ (or with true is always true)

$AB + C = C$ (or with true is always true)

So $C \implies ABC' + C = AB + C$

If $C$ is false

$ABC' + C$ = ($AB$ and true ) or false = $AB$ or false = $AB$

$AB + C = AB$ or false $= AB$

So $C' \implies ABC' + C = AB + C $

As either one of $C$ or $C'$ is true, $ ABC' + C = AB + C$