Simplify the Boolean function using k map. F(A,B,C,D) = √(0, 1, 2, 4, 5, 6, 8, 9, 13) a. What 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 Karnaugh Map, we need to follow these steps:

1. Create the K-map: A K-map for a 4-variable function will be a 4x4 grid representing all possible combinations of the variables A, B, C, and D.

2. Fill in the K-map: Populate the K-map with 1's in the cells corresponding to the minterms (the decimal numbers given in the square root notation, which represent the sum of minterms).

3. Identify Groups: Find the largest possible groups of 1's in the K-map. Each group must contain a number of 1's that is a power of 2 (e.g., 1, 2, 4, 8, etc.), and groups can wrap around the edges of the map.

4. Write the Simplified Function: For each group, write a product term (AND term) where a variable is included if it is the same within the entire group, and it is excluded if it varies. A variable is complemented if it is 0 within the group.

Let's go through these steps for the given function:

Step 1: Create the K-map

The K-map for a 4-variable function looks like this:

AB\CD	00	01	11	10
00
01
11
10

Step 2: Fill in the K-map

We fill in the K-map with 1's corresponding to the minterms √(0, 1, 2, 4, 5, 6, 8, 9, 13):

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

Step 3: Identify Groups

We can identify the following groups of 1's:

• A 8-cell group covering all cells in the first and second rows.
• A 2-cell group covering the cells in the second column of the third and fourth rows.

Step 4: Write the Simplified Function

For the 8-cell group (covering AB=00 and AB=01), the variables A and B are not consistent, so they are not part of the product term. C and D are both 0, so the product term is (\overline{C}\overline{D}).

For the 2-cell group (covering CD=01), the variable D is 1 and C is not consistent, so it is not part of the product term. A is 1 and B is 0, so the product term is (AB\overline{B}D).

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

[ F(A,B,C,D) = \overline{C}\overline{D} + A\overline{B}D ]

Boolean Algebra and Theorems

Boolean algebra is a branch of algebra that deals with variables that have two possible values: true (1) and false (0). It involves operations such as AND (conjunction), OR (disjunction), and NOT (negation).

Here are three important theorems of Boolean algebra:

1. Commutative Law

   • For AND: ( A \cdot B = B \cdot A )
   • For OR: ( A + B = B + A )

2. Associative Law

   • For AND: ( (A \cdot B) \cdot C = A \cdot (B \cdot C) )
   • For OR: ( (A + B) + C = A + (B + C) )

3. Distributive Law

   • ( A \cdot (B + C) = (A \cdot B) + (A \cdot C) )
   • ( A + (B \cdot C) = (A + B) \cdot (A + C) )

These laws and theorems are used to simplify Boolean expressions and functions, which is essential in the design and optimization of digital circuits.