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I have the the following logic function (where $'$ is NOT)

$f(a, b, c) = abc + ab'c + a'bc + a'b'c + ab'c'$

I have to simplify it as much as possible using only boolean algebra (no truth tables, Venn diagrams, etc.). So far I've managed to do the following

$= ab'(c'+c) + a'c(b+b') + abc$

$= ab' + a'c + abc$

I'm pretty sure it can be simplified further, but I'm not sure where to go from here.

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$$\begin{align} ab' + a'c + abc & = a(b' + bc) + a'c \\ & = a(b' + b)(b' + c) + a'c \\ & = a(b'+c) + a'c \\ & = ab'+ ac + a'c \\ & = ab' + (a+a')c \\ & = ab'+c \end{align}$$

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