Mastering Karnaugh Maps (K-Maps) for GCE A/L K-Maps | AL ICT | Unit 4 | Boolean Logic and Digital Circuit | in Tamil | தமிழில் English Medium

Mastering Karnaugh Maps (K-Maps) for GCE A/L

Welcome! If you are finding Boolean Algebra confusing, K-Maps are your best friend. They turn complex algebra into a visual puzzle. Here is everything you need to know, step-by-step.

1. What is the Purpose?

The main purpose of a K-Map is to simplify Boolean equations. Instead of using long algebraic laws (like De Morgan's or Distributive laws), we use a visual grid to group terms together and eliminate variables.

2. Truth Tables vs. K-Maps

• Normal Truth Tables (1D): These are lists. You read them from top to bottom. They show every possible input combination.
• K-Maps (2D): These are grids (tables). We take that 1D list and "fold" it into a 2D shape. This allows us to see patterns (neighbors) that are hard to see in a list.

3. SOP vs. POS

There are two ways to write equations, and two ways to use K-Maps:

• SOP (Sum of Products): You look for Minterms. In the K-Map, you place 1s and group the 1s.
• POS (Product of Sums): You look for Maxterms. In the K-Map, you place 0s and group the 0s.

Note: For this guide, we will focus on SOP (Grouping 1s) as it is the most common method for beginners.

4. Grid Sizes (Dimensions)

The size of your K-Map depends on the number of variables (inputs). For 3 Variables (x, y, z):

Total combinations = $2^3 = 8$.

You can arrange these 8 cells in different 2D shapes:

• 1 row × 8 columns (1x8)
• 8 rows × 1 column (8x1)
• 2 rows × 4 columns (2x4) (Most Common for 3 variables)
• 4 rows × 2 columns (4x2)

5. The Secret Weapon: Gray Code

This is the most important rule in K-Maps. When labeling the rows and columns, you cannot use normal binary counting (00, 01, 10, 11). You must use Gray Code.

Rule: Between any two adjacent numbers, only one bit (value) changes.

Sequence for 2 bits: 00 → 01 → 11 → 10

• 00 to 01: Only the right bit changed.
• 01 to 11: Only the left bit changed.
• 11 to 10: Only the right bit changed.

If you do not use Gray Code, your K-Map will not work!

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6. Step-by-Step Example

Let's solve this function together:

F(x,y,z) = (x'y'z) + (x'yz) + (xy'z) + (xyz') + (xyz)

Step 1: Convert to Binary (Minterms)

Look at each term. If a variable has a bar (like x'), it is 0. If it has no bar (like x), it is 1.

• x'y'z → 0 0 1 (Decimal 1)
• x'yz → 0 1 1 (Decimal 3)
• xy'z → 1 0 1 (Decimal 5)
• xyz' → 1 1 0 (Decimal 6)
• xyz → 1 1 1 (Decimal 7)

Step 2: Create the Truth Table

We list all 8 combinations (0 to 7). We put a 1 in the Output column if the number matches our list above (1, 3, 5, 6, 7). Otherwise, put a 0.

Decimal	x	y	z	Output (F)	Note
0	0	0	0	0
1	0	0	1	1	From x'y'z
2	0	1	0	0
3	0	1	1	1	From x'yz
4	1	0	0	0
5	1	0	1	1	From xy'z
6	1	1	0	1	From xyz'
7	1	1	1	1	From xyz

Step 3: Draw the K-Map Grid

We will use a 2 rows × 4 columns grid.

• Rows (x): 0, 1
• Columns (yz): 00, 01, 11, 10 (Remember Gray Code!)

x \ yz	00	01	11	10
0	0	1	1	0
1	0	1	1	1

We placed 1s in cells 1, 3, 5, 6, and 7 based on our Truth Table.

Step 4: Grouping (The Magic Step)

Rules for grouping:

1. Groups must contain $2^n$ cells (1, 2, 4, 8, 16...).
2. Groups must be rectangular or square.
3. Try to make groups as large as possible.
4. Every 1 must be inside at least one group.
5. Groups can overlap.

Let's group our example:

1. Group A (Red): Look at the middle two columns (01 and 11). We have four 1s forming a square (Cells 1, 3, 5, 7).
   Why? In this group, x changes (0 to 1) and y changes (0 to 1). But z is always 1.
   Result: z
2. Group B (Blue): Look at the bottom right corner. We have two 1s (Cells 6 and 7).
   Why? In this group, z changes (0 to 1). But x is always 1 and y is always 1.
   Result: xy

Step 5: Final Equation

Combine the results of the groups with an OR (+) sign.

F = z + xy

This is much simpler than the original long equation!

7. How to do POS (Product of Sums)

If the question asks for POS, or gives you Maxterms (0s):

1. Fill the K-Map with 0s instead of 1s (wherever the function is false).
2. Group the 0s together.
3. When writing the equation:
   • If a variable is 0 in the group, write it normally (e.g., A).
   • If a variable is 1 in the group, write it with a bar (e.g., A').
   • Combine variables with OR (+), and combine groups with AND (·).