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.