Question

minimize F(A,B,C)=A'BC+AB'C'+AB'C+AB+A'B' using karnaugh map

Answer 1

Minimization

• Minimization can be done using – Boolean algebra

BC + B C = B(C + C) = B • To combine terms

• Or equivalently – Karnaugh maps

• Visual identification of terms that can

be combined

Karnaugh Maps

• K-Maps are a convenient way to simplify Boolean

Expressions.

• They can be used for up to 4 (or 5) variables.

• They are a visual representation of a truth table.

• Expression are most commonly expressed in sum

of products form.

Answer 2

Given:

The Boolean function is $\[F\left( {A,B,C} \right) = A'BC + AB'C' + AB'C + AB + A'B'\]$.

To Find:

The minimized form of the given function using Karnaugh map.

Solution:

Firstly, we define the given function in its Canonical SOP (Sum of Products) form:

$\[\begin{array}{l}F\left( {A,B,C} \right) = A'BC + AB'C' + AB'C + AB + A'B'\\\\ \Rightarrow F\left( {A,B,C} \right) = A'BC + AB'C' + AB'C + AB\left( {C + C'} \right) + A'B'\left( {C + C'} \right)\\ \\\Rightarrow F\left( {A,B,C} \right) = A'BC + AB'C' + AB'C + ABC + ABC' + A'B'C + A'B'C'\end{array}\]$

The given function has three variables and hence $2^{3}=8$ cells K-map is required to minimize the expression.

We plot the K-map as shown in figure 1.

Now, we group the $1$s into three Quads in the K-map as shown in figure 2.

Lastly, we find the product terms by looking at the common variables present in each Quad and combining them to form a sum-of-product (SOP) form which yields the overall simplified Boolean expression.

Therefore, we obtain the simplified Boolean expression as:

$\[F\left( {A,B,C} \right) = B' + C + A\]$

Hence,  the minimized Boolean expression is $\[F\left( {A,B,C} \right) = B' + C + A\]$.

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