Question
Simplify the following expression (no double inversions)
to a sum of products (SOP).
Y = C * D * A * B
C d a and b have line on top
to a sum of products (SOP).
Y = C * D * A * B
C d a and b have line on top
Answers
Answered by
GPT-4o mini
To simplify the expression \( Y = C \cdot D \cdot A \cdot B \) where \( C, D, A, \) and \( B \) have negations (or complements) represented by a line on top, we first interpret the variables:
- \( \bar{C} \) for \( \overline{C} \) (C with a line on top)
- \( \bar{D} \) for \( \overline{D} \)
- \( \bar{A} \) for \( \overline{A} \)
- \( \bar{B} \) for \( \overline{B} \)
Thus, the expression given can be read as:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]
In Boolean algebra, a product of terms (also known as a product term or conjunction) is already in Sum of Products (SOP) form when we don't have double inversions (like \( \overline{\overline{x}} = x \)).
Since we want to express it as a Sum of Products and we have only one term in this case, it is already in the correct form.
To rephrase, the expression in SOP is simply:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]
This is the simplest Sum of Products form based on the information given. If you need to consider combinations of the negated and non-negated variables (which isn’t applicable here since there are only complements), we could derive Minterms based on combinations. However, as per your request, the given expression remains as is.
Thus, the final answer is:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]
- \( \bar{C} \) for \( \overline{C} \) (C with a line on top)
- \( \bar{D} \) for \( \overline{D} \)
- \( \bar{A} \) for \( \overline{A} \)
- \( \bar{B} \) for \( \overline{B} \)
Thus, the expression given can be read as:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]
In Boolean algebra, a product of terms (also known as a product term or conjunction) is already in Sum of Products (SOP) form when we don't have double inversions (like \( \overline{\overline{x}} = x \)).
Since we want to express it as a Sum of Products and we have only one term in this case, it is already in the correct form.
To rephrase, the expression in SOP is simply:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]
This is the simplest Sum of Products form based on the information given. If you need to consider combinations of the negated and non-negated variables (which isn’t applicable here since there are only complements), we could derive Minterms based on combinations. However, as per your request, the given expression remains as is.
Thus, the final answer is:
\[ Y = \bar{C} \cdot \bar{D} \cdot \bar{A} \cdot \bar{B} \]