Simplify expression Boolean-Algebra

Question

For my class I have to study some Boolean algebra. Now I'm having some difficulties with simplifying expression.

For example I get:

A.B.C + NOT(A) + NOT(B) + NOT(C)

I tried checking wolfram alpha but there's not simplification showing up there. Can you tell me how to simplify this expression?

Thanks in advance

Answer 1

Boolean Algebraic Solution (using a more traditional notation):

Given Boolean expression:

abc + a' + b' + c'

Apply double negation:

(abc + a' + b' + c')''

Apply De Morgan's Law for a disjunction:

((abc)'a''b''c'')'

Reduce double negations:

((abc)'abc)'

AND of x and x' is 0:

(0)'

Negation of 0 is 1:

1

Boolean Algebraic Solution (using the given notation):

Given Boolean expression:

a.b.c + NOT(a) + NOT(b) + NOT(c)

Apply double negation:

NOT(NOT(a.b.c + NOT(a) + NOT(b) + NOT(c)))

Apply De Morgan's Law for a disjunction:

NOT(NOT(a.b.c).NOT(NOT(a)).NOT(NOT(b)).NOT(NOT(c))))

Reduce double negations:

NOT(NOT(a.b.c).a.b.c)

AND of x and x' is 0:

NOT(0)

Negation of 0 is 1:

1

Answer 2

Truth table:

A    B    C    X
0    0    0    1
0    0    1    1
0    1    0    1
0    1    1    1
1    0    0    1
1    0    1    1
1    1    0    1
1    1    1    1

So the simplification is just:

X = 1

Answer 3

Wolfram Alpha wasn't giving a simplification because it didn't understand your notation. Using (A and B and C) or NOT(A) or NOT(B) or NOT(C) shows that it simplifies to true.

Or you can just look at it: if any are false, the NOT will makke everything true, and if they're all true, then so is the first clause.

Answer 4

Use http://en.wikipedia.org/wiki/De_Morgan%27s_laws

A AND B = NOT(NOT(A) OR NOT(B))

A OR B = NOT(NOT(A) AND NOT(B))

and the normal methods for distribution, commutation, etc. See http://en.wikipedia.org/wiki/Boolean_algebra#Laws

Note that in the linked texts above the symbols are written differently.

• AND is ∧
• OR is ∨
• NOT is ¬