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how do i simplify this equation using boolean algebra:

AB + ¬AC + BC

to be equal to

AB + ¬AC

the BC is unneeded, but how do i remove that term using boolean algebra?

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2 Answers 2

$$\begin{align*} AB+\lnot AC+BC&=AB+\lnot AC+(A+\lnot A)BC\\ &=AB+\lnot AC+ABC+\lnot ABC\\ &=(AB+ABC)+(\lnot AC+\lnot ACB)\\ &=AB+\lnot AC \end{align*}$$

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We have: $$ A + \lnot A = 1 $$

And: $$ BC = 1 \cdot BC = (A + \lnot A)BC = ABC + \lnot ABC $$

Plug into your expression to get: \begin{align*} AB + \lnot AC + BC &= AB + \lnot AC + (ABC + \lnot ABC) \\ &= (AB + ABC) + (\lnot AC + \lnot ABC) \\ &= \left(AB + (AB)C\right) + \left(\lnot AC + (\lnot AC)B\right) \\ &= AB + \lnot AC \end{align*}

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