Boolean Logic Using Proofs
$\Begingroup$ 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...
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.
2 Answers
$$AB \overline C + C$$ 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: $X + \overline X = 1$ $$AB + C$$ $$AB \overline C + C = AB + C$$
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$
David Miller brings 15 years of experience in global economics, personal finance strategy, and market dynamics. He specializes in turning complex economic trends into actionable insights for everyday readers.
