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boolean-logicsimplification

Boolean Simplification

I have a boolean simplification problem that's already been solved.. but I'm having a hard time understanding one basic thing about it.. the order in which it was solved.

The problem is simplifying this equation:

Y = ¬A¬B¬C + ¬AB¬C + A¬B¬C + A¬BC + ABC

The solution is:

Y = ¬A¬B¬C + ¬AB¬C + A¬B¬C + A¬BC + ABC
  = ¬A¬B¬C + ¬AB¬C + A¬B¬C + A¬BC + A¬BC + ABC (idempotency for A¬BC) 
  = ¬A¬C(¬B + B) + A¬B(¬C + C) + AC(¬B + B)
  = ¬A¬C + A¬B + AC

The way I solved it is:

Y = ¬A¬B¬C + ¬AB¬C + A¬B¬C + A¬BC + ABC
  = ¬A¬B¬C + ¬AB¬C + ¬A¬B¬C + A¬B¬C + A¬BC + ABC (idempotency for ¬A¬B¬C)
  = ¬A¬C(¬B + B) + ¬B¬C(¬A + A) + AC(¬B +B)
  = ¬A¬C + ¬B¬C + AC

So how do I know which term to use the law of idempotency on? Thanks.

Solution

  • ¬A¬B¬C + ¬AB¬C + A¬B¬C + A¬BC + ABC

    ¬A¬C(¬B + B) + A(¬B¬C + ¬BC + BC)

    ¬A¬C + A(¬B¬C + ¬BC + BC) <- see truth table below for the simplification of this

    ¬A¬C + A(¬B + C)

    ¬A¬C + A¬B + AC

    truth table:

    B C

    0 0 = 1 + 0 + 0 = 1

    0 1 = 0 + 1 + 0 = 1

    1 0 = 0 + 0 + 0 = 0

    1 1 = 0 + 0 + 1 = 1

    which is ¬B + C