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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.

Jonathan
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  • The simplest way is to use a truth table. If you insist on writing sentences, assume the left-hand side is true and argue that the right-hand side must also be true and then assume the right-hand side is true and prove that the left-hand side must be true. – John Douma Apr 07 '16 at 20:28
  • My question is exactly that. The truth table is obviously very self explanatory for this expression. But how would the proof look like? – Jonathan Apr 07 '16 at 21:04
  • The truth table is the proof. If two logical expressions have the same truth table they are equivalent. If you find that approach to be distasteful then follow the recipe above. Suppose $ABC' + C$ is true ... – John Douma Apr 07 '16 at 21:07

2 Answers2

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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$

sanketalekar
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$$AB \overline C + C$$ 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: $X + \overline X = 1$ $$AB + C$$ $$AB \overline C + C = AB + C$$

StainlessSteelRat
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