Following the K-MAP of a boolean function of five variables A, B, C, D and E. The minimum SOP is _____.

To determine the minimum Sum of Products (SOP) from a Karnaugh Map (K-MAP) for a boolean function with five variables $A$, $B$, $C$, $D$, and $E$, follow these steps:

1. Break Down the Problem

You need to:

1. Analyze the K-MAP for the five variables.
2. Identify and group the 1's to create the simplest SOP expression.

2. Relevant Concepts

The K-MAP allows for visual simplification of boolean expressions. Grouping rules include:

• Groups can be of size $1$, $2$, $4$, $8$, etc. (i.e., powers of two).
• Each group should be as large as possible, covering adjacent 1's.
• Groups can wrap around the edges of the K-MAP.

3. Analysis and Detail

• Identify all the groups of 1's in the K-MAP.
• Write the product term for each group, reducing the number of variables by excluding those that change within the group.

4. Verify and Summarize

Check that:

• Each combination of variables in your SOP covers all the 1's in the K-MAP.
• There may be several groupings; ensure you choose the one that yields the minimum SOP expression.

Final Answer

Since I don’t have a visual representation of the K-MAP in your question, I cannot provide the exact minimum SOP expression. Please refer to the identified groups in your K-MAP and apply the grouping rules outlined above to arrive at your solution.