Boolean Expression Simplification Process
The goal is to simplify the given boolean expression to its minimal form. The expression provided is:
a′b′+ab+a′ba′b′+ab+a′b
We can simplify this expression using the rules of boolean algebra. Let's break down the simplification step-by-step:
Step 1: Applying Boolean Algebra Rules
- Consider the terms a′b′a′b′ and a′ba′b. We can factor out a′a′ using the distributive law: XY+XZ=X(Y+Z)XY+XZ=X(Y+Z).
- Applying this, we get: a′b′+a′b=a′(b′+b)a′b′+a′b=a′(b′+b).
- We know that for any boolean variable xx, x+x′=1x+x′=1. Therefore, b′+b=1b′+b=1.
- Substituting this back, we have a′(1)a′(1).
- Since X⋅1=XX⋅1=X, a′(1)a′(1) simplifies to a′a′.
Step 2: Substituting Simplified Terms
Now, substitute the simplified term (a′a′) back into the original expression:
Original expression: a′b′+ab+a′ba′b′+ab+a′b
After simplifying a′b′+a′ba′b′+a′b, the expression becomes:
a′+aba′+ab
Step 3: Final Simplification
We need to simplify the expression a′+aba′+ab.
There is a useful boolean algebra identity: X′+XY=X′+YX′+XY=X′+Y.
In our expression, let X=aX=a and Y=bY=b. Applying the identity:
a′+ab=a′+ba′+ab=a′+b
This is the minimal form of the given boolean expression.
Conclusion
The simplified form of the boolean expression a′b′+ab+a′ba′b′+ab+a′b is a′+ba′+b. This matches option 2.
