[Solved]: Combinational Logic – Need help with proof by rewrite

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Problem Detail: In my Computer Logic class we were assigned the following problem:

Complete a truth table that has $3$ inputs $(A, B,C)$ and one output $(F)$. $F$ is asserted whenever $B$ or $C$ are asserted, but deasserted if both $B$ and $C$ are asserted. $F$ is asserted whenever $A$ is asserted and $C$ is deasserted. $F$ is deasserted in all other cases.

After which we are supposed to

Determine the sum-of-products equation for $F$. Show the terms instead of the minterm abbreviations (show $ABC$ instead of m7).

Finally, we are supposed to:

Prove that $F$ from question $2$ is equivalent to $F = BC’ + B’C + AC’$ using proof by rewrite. HINT: remember the uniting law is the key to minimization.

Here is what I have: 1) 

A    B    C    F 0    0    0    0 0    0    1    1 0    1    0    1 0    1    1    0 1    0    0    1 1    0    1    1 1    1    0    1 1    1    1    1

2) $F = A’B’C + A’BC’ + AB’C’ + AB’C + ABC’$ Number three is what is throwing me. I am using the axioms of boolean algebra, such as the Uniting Law, but so far all I have been able to do is the following: 3) A’B’C + A’BC’ + AB’C’ + AB’C + ABC’ = BC’ + B’C + AC’ 

A'B'C + A'BC' + AB'C' + AB'C + ABC'  A'B'C + AB'C + A'BC' + AB'C'  + ABC'            Commutative Law  B'C + A'BC' + AB'C' + ABC'                      Uniting Law  A'BC' + ABC' + B'C + AB'C'                      Commutative Law  BC' + B'C + AB'C'                               Uniting Law

That’s as far as I am able to get. Did I screw up the truth table, resulting in incorrect minterms, or did I miss something? Appreciate the help!

Asked By : allCrocs

Answered By : TEMLIB

The last line is wrong : 1 1 1 1 Should be : 1 1 1 0 But the equations following are correct. For the simplification : BC’ + B’C + AB’C’ = BC’ + B’.(C+AC’) = BC’+B’.(C+A)=BC’+B’C+B’A (I don’t know the name of the rules used, there is some associativity, “X+X’Y = X+Y”, etc…) — edit — Let’s prove that X+X’Y = X+Y Using :

  • A = A.B + A.B’
  • A = A+A
  • A + AB = A.(1+B) = A
  • A + A’ = 1

X+X’Y = XY+XY’+X’Y = Y(X+X’) + XY’ = Y + XY’ Repeating (A=A+A) : X+X’Y = X+X’Y + X+X’Y = X+X’Y + Y+XY’ = X + Y + X’Y + XY’ = X+XY’ + Y+X’Y = X + Y

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/38025