Question

using consensus theorem, we could eliminate at least two terms from the following logic expression: $abc + bcd + acd + abd + bcd + acd$

Explanation:

Step1: Recall Consensus theorem

The Consensus theorem states that $XY + X'Z+YZ=XY + X'Z$. We need to find terms in the given expression $A'B'C + BC'D' + A'CD + AB'D' + BCD + AC'D'$ that fit the pattern of the Consensus - theorem.

Step2: Identify pairs

Let's consider the terms:

• For the terms $A'B'C$ and $A'CD$, the consensus term is $A'BD$. But this term is not in the original expression.
• Consider $BC'D'$ and $BCD$. The consensus term is $BD'$. Not in the original expression.
• Consider $BC'D'$ and $AC'D'$. Here $X = B$, $X'=A$, $Y = C'D'$, $Z = C'D'$. The consensus term is $C'D'$ which is not in the original expression.
• Consider $A'CD$ and $BCD$. The consensus term is $CD$. Not in the original expression.
• Consider $AB'D'$ and $AC'D'$. The consensus term is $AD'$. Not in the original expression.
• Now, consider the terms $A'B'C$ and $AB'D'$. The consensus term is $B'CD'$. Not in the original expression.
• Consider the terms $A'CD$ and $AB'D'$. The consensus term is $B'CD$. Not in the original expression.
• Consider the terms $BC'D'$ and $A'B'C$. The consensus term is $A'B' D'$. Not in the original expression.
• However, consider the terms $A'B'C+A'CD + BCD$.
• Here, if we consider $X = A'C$, $Y = B'$, $Z = D$, then $A'B'C+A'CD=A'C(B' + D)$ and the consensus term of $A'B'C$ and $A'CD$ with respect to the variable - set is $BCD$ (by Consensus theorem: $A'B'C+A'CD + BCD=A'B'C+A'CD$).
• Also, consider $BC'D'+AB'D'+AC'D'$. Here, if $X = B C'$, $X'=A$, $Y = D'$, $Z = C'D'$, the consensus term of $BC'D'$ and $AB'D'$ is $AC'D'$. So $BC'D'+AB'D'+AC'D'=BC'D'+AB'D'$.

Step3: Simplify the expression

The original expression $A'B'C + BC'D' + A'CD + AB'D' + BCD + AC'D'$ simplifies to $A'B'C+A'CD + BC'D'+AB'D'$.

Answer:

$A'B'C+A'CD + BC'D'+AB'D'$