Logical statements: $A+B+C=A+B+C+AB+BC+AC$

Question

Let $A,B, C$ be logical statements. Then: $A+B+C=A+B+C+AB+BC+AC$

Prove it without a table.

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My attempt:

$$A=A\cdot(1+B+C)$$ $$B=B\cdot(1+A+C)$$ $$C=C\cdot(1+A+B)$$ $$\implies A+B+C= A+AB+AC+B+AB+BC+C+BC+AC$$ $$\implies A+B+C=A+B+C+AB+AB+BC+BC+AC+AC$$ $$\implies A+B+C=A+B+C+AB+BC+AC$$

Is this correct?

Answer 1

Yes. it is completely correct.

You could have shortened it a bit by writing

$$A+B+C+AB+BC+AC = A(1+B+C) + B(1+C) + C $$ $$ \stackrel{1+x=1,1x=x}{=}A+B+C$$

Answer 2

It's correct, if you are using Monotone laws of Boolean algebra or Logical equivalence, we can write$:$ \begin{align} &\hspace{3ex}A+B+C\\ &=A+AB+B+BC+C+CA\tag*{Absorption law}\\ &=A+B+C+AB+BC+AC\tag*{Commutative law} \end{align}