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This image shows the solution of a Boolean function using Karnaugh maps (K-map). Here’s a simplified explanation:

### 1. Boolean Function
The function is given as:
\[ f(x, y, z) = xy + yz + xz \]

This is a Boolean expression in terms of three variables (x, y, and z). It is shown in sum-of-products (SOP) form, where terms like \( xy \), \( yz \), and \( xz \) represent AND operations between the variables.

### 2. K-map for Simplification
- A K-map is a grid used to simplify Boolean expressions. It helps identify patterns like groups of 1s (which can be simplified).
- The grid you see represents the possible combinations of the variables \( x \), \( y \), and \( z \). In this case, the variables are represented by two sets of columns, one for \( BC \) and one for \( AB \).

### 3. Grouping and Simplification
- The K-map highlights groups of 1s, and you can group them to simplify the expression.
- For example, there’s a group formed by the 1s in the columns and rows. The group simplification reduces the Boolean expression to simpler terms.
- After grouping, the expression becomes simplified as:
\[ D + BC' \]
where:
  - \( D \) represents one of the groups from the K-map.
  - \( BC' \) represents another simplified group.

### 4. Final Simplified Expression
The final simplified Boolean expression using K-map is:
\[ D + BC' \]

In simpler terms:
- \( D \) is a term derived from a group of 1s, and \( BC' \) is another term from the K-map simplification process.

### Summary:
1. You start with a complex Boolean function.
2. Then, you use the K-map to simplify the function by grouping 1s.
3. The final expression becomes simpler, like \( D + BC' \), which is easier to implement in digital circuits.

Would you like to dive deeper into any of the steps or need further clarification? 

Here are some follow-up questions for further learning:
1. How do you fill in a K-map?
2. What are the rules for grouping 1s in a K-map?
3. What is the significance of the SOP (sum-of-products) form in Boolean algebra?
4. Can you solve another example with a different Boolean function?
5. How does the K-map help in minimizing logical expressions?

**Tip:** When grouping in a K-map, always aim to make the groups as large as possible (powers of 2) for the simplest result.

Learn how to simplify Boolean functions using Karnaugh maps with a step-by-step explanation. This example demonstrates the simplification of the Boolean function f(x, y, z) = xy + yz + xz, resulting in the simplified expression D + BC'. Suitable for high school students (Grades 10-12).

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