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Boolean algebra simplification to lowest form

I am relearning Boolean algebra for a class, but I cant seem to simplify this expression any further. It is possible that its fully simplified but I wanted a second opinion.

the expression:

(!a*!b*!c)+(!c*(b*a))+(!a*(c*b)) where * is and ; + is or
Solution

  • First:

    (!a * !b * !c) == !(a + b + c)

    Second:

    (!c * (b * a)) == (!c * b * a)

    Now the second and third ORed sections can be simplified to:

    b * a ^ c

    Where ^ is an XOR (exclusive OR: either a or c, but not both). If you take the order of precedence of operators into account, you may be able to discard even more parentheses, but to play it safe:

    !(a + b + c) || (b * a ^ c)