Question

Find a minimal sum-of-products expansion, given the \({\bf{K}}\)-map shown with don't care conditions indicated with \({\bf{d}}'{\bf{s}}\).

Short answer

Minimal sum-of-products expansion \({\bf{\bar z + wx}}\)

Step-by-step solution

Step 1:Definition

In some circuits we care only about the output for some combinations of input values, because other combinations of input values are not possible or never occur. This gives us freedom in producing a simple circuit with the desired output because the output values for all those combinations that never occur can be arbitrarily chosen. The values of the function for these combinations are called don’t care conditions.

Largest block

Given:

The largest block in the graph consists of two columns \({\bf{\bar yz}}\) and \({\bf{\bar y\bar z}}\) (as these two columns contain only \(1{\bf{'s}}\)and \({\bf{d's}}\)), while the two columns can be notated as \({\bf{\bar z}}\) (as the two columns contain the only possible outcomes with \({\bf{\bar z}}\) ).

Only the element \({\bf{wx \bar yz}}\) was not included in the previous block, we still need to include it in a block. The largest block that contains \({\bf{wx\bar yz}}\) is the row \({\bf{wx}}\) (as the entire row contains \({\bf{d's}}\)and \(1{\bf{'s}}\)).

Minimal sum-of-products

The minimal sum-of-products expansion is then the sum of these blocks.

Hence, the Minimal sum-of-products expansion \({\bf{\bar z + wx}}\).