Given a Boolean function F(A, B, C) = Σ(0, 1, 2, 3, 5), what is the expression in SOP form?
A'B'C' + A'B'C + A'BC + A'BC' + AB'C
Understanding the Boolean Function F(A, B, C) and its SOP Form
The question asks us to find the Sum of Products (SOP) form for the given Boolean function F(A, B, C). The function is specified using the $\Sigma$ notation, F(A, B, C) = $\Sigma$(0, 1, 2, 3, 5). This notation means that the function F(A, B, C) outputs a logic '1' for the minterms corresponding to the decimal indices 0, 1, 2, 3, and 5.
What is Sum of Products (SOP) Form?
The Sum of Products (SOP) form is a way to express a Boolean function as a sum of product terms. Each product term represents one combination of input variables for which the function output is '1'. These product terms are called minterms when they include all input variables, either in their original or complemented form.
Deriving Minterms from $\Sigma$(0, 1, 2, 3, 5)
For a Boolean function with three variables (A, B, C), there are $2^3 = 8$ possible combinations of inputs, represented by minterms $m_0$ through $m_7$. The $\Sigma$ notation lists the decimal indices of the minterms for which the function is true (outputs '1'). We need to find the minterm expression for each given index:
| Decimal Index | Binary Representation (A B C) | Minterm |
|---|---|---|
| 0 | 0 0 0 | A'B'C' ($m_0$) |
| 1 | 0 0 1 | A'B'C ($m_1$) |
| 2 | 0 1 0 | A'BC' ($m_2$) |
| 3 | 0 1 1 | A'BC ($m_3$) |
| 5 | 1 0 1 | AB'C ($m_5$) |
Remember that for a minterm:
- If a variable's value is 0 in the binary representation, it appears as its complement (e.g., A').
- If a variable's value is 1, it appears in its original form (e.g., A).
Constructing the SOP Expression for the Boolean function F(A, B, C)
The Sum of Products (SOP) form of the Boolean function F(A, B, C) is obtained by summing (ORing) the minterms that correspond to the decimal indices listed in the $\Sigma$ notation. In this case, the indices are 0, 1, 2, 3, and 5.
So, the SOP form of the Boolean function F(A, B, C) is the sum of $m_0$, $m_1$, $m_2$, $m_3$, and $m_5$.
F(A, B, C) = $m_0 + m_1 + m_2 + m_3 + m_5$
Substituting the minterm expressions we found:
F(A, B, C) = A'B'C' + A'B'C + A'BC' + A'BC + AB'C
This is the required SOP form for the given Boolean function F(A, B, C).
Important Questions from Minimization of Boolean Expression
What is the value of \( \bar{F}\)?
\(F = AB + \bar{C}\bar{D} + \bar{B}D\)
Simplify the following Boolean expression.
E(E + F) + DE + D(E + F)
Which statement(s) is/are correct regarding the Boolean algebra?
I. It facilitate the analysis and design of digital circuits.
II. Expresses in algebraic form the input-output relationship of logic diagram.
The input-output relationship of the binary variable for each gate can be represented in tabular form by a _______.
What is the simplified expression for the Boolean function F(A, B, C, D) = Σ(0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14) using the K - map method?
