Expand the following expression P(C,A,B,D) = ABC into mini terms.

Steps to Solve

1. Identify the Variables The expression given is $P(C, A, B, D) = ABC$. The variables involved are $A$, $B$, $C$, and $D$.

2. Understand the Mini-Terms Mini-terms are products (AND expressions) of all variables in the expression, each either in true or complemented form. This means we need to consider all possible combinations of the variables to express them.

3. Determine the Full Set of Mini-Terms For four variables ($C$, $A$, $B$, and $D$), there are $2^4 = 16$ possible combinations. We can represent them as follows: $$ \text{Mini-terms} = A'B'C'D', A'B'C'D, A'B'CD', A'B'CD, A'BC'D', A'BC'D, A'BCD', A'BCD, ABC'D', ABC'D, ABCD', ABCD $$

4. Express the Given Function in Terms of Mini-Terms The expression $ABC$ can be expanded to include all combinations of the remaining variable $D$.
   In this case, we have: $$ P(C, A, B, D) = ABCD + ABCD' $$ This expression incorporates both scenarios where $D$ is true and where $D$ is false.

5. Write the Final Expanded Expression Thus, the final expression in mini-terms is: $$ P(C, A, B, D) = m_{5} + m_{7} $$ where $m_5 = ABCD$ and $m_7 = ABCD'$.