Simplify the Boolean function using k map. F(A,B,C,D) = √(0, 1, 2, 5, 6, 8, 9, 13, 14) a) Is Boolean algebra write any three theorems of Boolean algebra.

Simplifying the Boolean Function using Karnaugh Map (K-map)

To simplify the given Boolean function using a K-map, we first need to understand that the function is represented using a min-term notation, where √(0, 1, 2, 5, 6, 8, 9, 13, 14) indicates the square root of the sum of min-terms. However, in Boolean algebra, this is typically represented as Σ (sum of min-terms) rather than √. So, the correct representation of the function is:

F(A, B, C, D) = Σ(0, 1, 2, 5, 6, 8, 9, 13, 14)

Now, let's simplify this function using a 4-variable K-map:

Step 1: Set up the K-map

A 4-variable K-map is a 4x4 grid, with each cell representing a unique combination of the variables A, B, C, and D. The rows and columns are ordered in such a way to minimize the difference between adjacent cells (Gray code).

AB\CD	00	01	11	10
00
01
11
10

Step 2: Populate the K-map with the given min-terms

We place a '1' in the cells corresponding to the min-terms given in the function:

AB\CD	00	01	11	10
00	1	1		1
01			1	1
11		1		1
10	1	1

Step 3: Group the ones in the K-map

We group the ones in the K-map into the largest possible groups of 1, 2, 4, 8, or 16. Each group must be a power of two and the groups can wrap around the edges of the map.

Here are the groups:

• Group 1: Covers min-terms 0, 1, 8, 9 (covers the top two cells in the first and second columns)
• Group 2: Covers min-terms 2, 6, 10, 14 (covers the top right cell and bottom right cell in the first and fourth columns)
• Group 3: Covers min-terms 5, 13 (covers the middle cells in the third column)

Step 4: Write the simplified Boolean expression

For each group, we write a product term (AND) where a variable is included if it is the same for all min-terms in the group, and excluded if it varies.

• Group 1: A'B' (since A and B are 0 for all min-terms in the group)
• Group 2: CD' (since C is 1 and D is 0 for all min-terms in the group)
• Group 3: BC'D (since B is 1, C is 0, and D is 1 for all min-terms in the group)

The simplified Boolean function is the sum (OR) of these product terms:

F(A, B, C, D) = A'B' + CD' + BC'D

Boolean Algebra Theorems

Here are three important theorems of Boolean algebra:

1. Identity Law:

   • A + 0 = A
   • A . 1 = A

2. Complement Law:

   • A + A' = 1
   • A . A' = 0

3. Distributive Law:

   • A + (B . C) = (A + B) . (A + C)
   • A . (B + C) = (A . B) + (A . C)

Examples:

1. Identity Law Example:

   • Let A = 1, then A + 0 = 1 + 0 = 1
   • Let A = 0, then A . 1 = 0 . 1 = 0

2. Complement Law Example:

   • Let A = 1, then A + A' = 1 + 0 = 1
   • Let A = 0, then A . A' = 0 . 1 = 0

3. Distributive Law Example:

   • Let A = 1, B = 0, C = 1, then A + (B . C) = 1 + (0 . 1) = 1 + 0 = 1 and (A + B) . (A + C) = (1 + 0) . (1 + 1) = 1 . 1 = 1
   • Let A = 1, B = 0, C = 1, then A . (B + C) = 1 . (0 + 1) = 1 . 1 = 1 and (A . B) + (A . C) = (1 . 0) + (1 . 1) = 0 + 1 = 1

These theorems are fundamental to simplifying Boolean expressions and designing digital logic circuits.