Simplify the following Boolean expression. E(E + F) + DE + D(E + F)
E + DF
Simplifying Boolean Expression E(E + F) + DE + D(E + F)
Let's simplify the given Boolean expression step by step using fundamental Boolean algebra laws.
The expression we need to simplify is: \(\small E(E + F) + DE + D(E + F)\)
Step-by-Step Simplification
We will apply the distributive law and other common Boolean identities to simplify the expression.
- First, let's expand the terms using the distributive law, \(A(B + C) = AB + AC\):
- \(\small E(E + F) = EE + EF\)
- \(\small D(E + F) = DE + DF\)
- Now, substitute these back into the original expression:
\(\small (EE + EF) + DE + (DE + DF)\)
- Apply the Idempotent Law, \(A \cdot A = A\), to the term \(\small EE\):
\(\small EE = E\)
The expression becomes: \(\small (E + EF) + DE + (DE + DF)\)
- Apply the Absorption Law, \(A + AB = A\), to the term \(\small E + EF\):
\(\small E + EF = E\)
The expression becomes: \(\small E + DE + DE + DF\)
- Apply the Idempotent Law, \(A + A = A\), to the term \(\small DE + DE\):
\(\small DE + DE = DE\)
The expression becomes: \(\small E + DE + DF\)
- Apply the Absorption Law, \(A + AB = A\), to the term \(\small E + DE\):
\(\small E + DE = E\)
The final simplified expression is: \(\small E + DF\)
Summary of Simplification Steps
Starting from the original expression, we applied Boolean algebra rules:
\(\small E(E + F) + DE + D(E + F)\)
Distribute E and D:
\(\small (EE + EF) + DE + (DE + DF)\)
Idempotent Law (\(\small EE = E\)):
\(\small (E + EF) + DE + DE + DF\)
Absorption Law (\(\small E + EF = E\)):
\(\small E + DE + DE + DF\)
Idempotent Law (\(\small DE + DE = DE\)):
\(\small E + DE + DF\)
Absorption Law (\(\small E + DE = E\)):
\(\small E + DF\)
The simplified Boolean expression is \(\small E + DF\).
Revision Table: Key Boolean Algebra Laws
| Law Name | AND (\(\cdot\)) Form | OR (+) Form |
|---|---|---|
| Identity Law | \(\small A \cdot 1 = A\) | \(\small A + 0 = A\) |
| Null Law | \(\small A \cdot 0 = 0\) | \(\small A + 1 = 1\) |
| Idempotent Law | \(\small A \cdot A = A\) | \(\small A + A = A\) |
| Involution Law | \(\small (A')' = A\) | - |
| Complementary Law | \(\small A \cdot A' = 0\) | \(\small A + A' = 1\) |
| Commutative Law | \(\small A \cdot B = B \cdot A\) | \(\small A + B = B + A\) |
| Associative Law | \(\small A \cdot (B \cdot C) = (A \cdot B) \cdot C\) | \(\small A + (B + C) = (A + B) + C\) |
| Distributive Law | \(\small A \cdot (B + C) = (A \cdot B) + (A \cdot C)\) | \(\small A + (B \cdot C) = (A + B) \cdot (A + C)\) |
| Absorption Law | \(\small A \cdot (A + B) = A\) | \(\small A + (A \cdot B) = A\) |
Additional Information on Boolean Expression Simplification
Simplifying Boolean expressions is crucial in digital logic design for reducing the number of logic gates required, which leads to simpler circuits, lower cost, reduced power consumption, and faster operation. Besides algebraic simplification using laws, Karnaugh maps (K-maps) and the Quine-McCluskey method are also used, especially for expressions with more variables.
- Karnaugh Maps: A graphical method suitable for expressions up to 5 or 6 variables. It helps visualize adjacent terms that can be combined.
- Quine-McCluskey Method: A tabular method suitable for automating simplification, especially for expressions with a large number of variables.
- Importance: Simplified expressions directly translate to optimized logic gates like AND, OR, and NOT gates, forming the basis of digital circuits.
Mastering Boolean algebra laws is fundamental before moving to more advanced simplification techniques.
Important Questions from Minimization of Boolean Expression
Consider the following four variable Boolean function in sum-of-product form
$F(b_3, b_2, b_1, b_0) = \sum(0, 2, 4, 8, 10, 11, 12)$.
where the value of the function is computed by considering $b_3b_2b_1b_0$ as a 4-bit binary number, where $b_3$ denotes the most significant bit and $b_0$ denotes the least significant bit. Note that there are no don't care terms. Which ONE of the following options is the CORRECT minimized Boolean expression for F?
- Simplified Boolean equation for the following truth table is :
x y z F 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 The simplified form of a Boolean equation $(A\overline{B} + A\overline{B}C + AC)(\overline{A}\overline{C} + \overline{B})$ is :
What is the value of \( \bar{F}\)?
\(F = AB + \bar{C}\bar{D} + \bar{B}D\)
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.
