Question
Simplify the following expression (no double inversions)
to a sum of products (SOP).
Y = C * D * A * B
A and B have line on top inside of C and D
to a sum of products (SOP).
Y = C * D * A * B
A and B have line on top inside of C and D
Answers
Answered by
GPT-4o mini
To simplify the expression \( Y = C \cdot D \cdot \overline{A} \cdot \overline{B} \) into a sum of products (SOP) form, we have the following variables:
- \( C \)
- \( D \)
- \( \overline{A} \) (the inversion of \( A \))
- \( \overline{B} \) (the inversion of \( B \))
The given expression is already in a product form where the product consists of four literals. In Boolean algebra, a product of literals is already considered one of the simplest forms (as it is a single term).
However, for clarity and to help visualize the components of the SOP, we can represent \( Y \) as:
\[
Y = C \cdot D \cdot \overline{A} \cdot \overline{B}
\]
Since this expression is a single product term, it can also be considered a "minterm" for the state where \( C \) and \( D \) are true, and \( A \) and \( B \) are false.
Thus, there is no further simplification needed as there are no combinations to create as we are dealing with a single product term.
In conclusion, the simplest form of the expression \( Y \) is already:
\[
Y = C \cdot D \cdot \overline{A} \cdot \overline{B}
\]
If you'd like to see it in a more traditional SOP format, you can note that it contains a single product term:
\[
Y = 1 \cdot (C \cdot D \cdot \overline{A} \cdot \overline{B})
\]
This essentially confirms that it is in SOP form, although typically we talk about sums of multiple product terms when discussing SOP. Here, it remains as one product term.
- \( C \)
- \( D \)
- \( \overline{A} \) (the inversion of \( A \))
- \( \overline{B} \) (the inversion of \( B \))
The given expression is already in a product form where the product consists of four literals. In Boolean algebra, a product of literals is already considered one of the simplest forms (as it is a single term).
However, for clarity and to help visualize the components of the SOP, we can represent \( Y \) as:
\[
Y = C \cdot D \cdot \overline{A} \cdot \overline{B}
\]
Since this expression is a single product term, it can also be considered a "minterm" for the state where \( C \) and \( D \) are true, and \( A \) and \( B \) are false.
Thus, there is no further simplification needed as there are no combinations to create as we are dealing with a single product term.
In conclusion, the simplest form of the expression \( Y \) is already:
\[
Y = C \cdot D \cdot \overline{A} \cdot \overline{B}
\]
If you'd like to see it in a more traditional SOP format, you can note that it contains a single product term:
\[
Y = 1 \cdot (C \cdot D \cdot \overline{A} \cdot \overline{B})
\]
This essentially confirms that it is in SOP form, although typically we talk about sums of multiple product terms when discussing SOP. Here, it remains as one product term.