Simplify the Boolean function using K-map. F(x,y,z) = √ (0,1,2,4,5,6,8,9,12,13,14)

To simplify the given Boolean function using a Karnaugh map (K-map), we will follow these steps:

Step 1: Set up the K-map

A K-map for a three-variable function will have 2^3 = 8 cells, arranged in a 2x4 grid. Each cell represents a minterm or a combination of the variables x, y, and z.

Here is the layout of the K-map with the variables assigned to rows and columns:

      yz
      00  01  11  10
   -----------------
x=0 |    |    |    |    |
   -----------------
x=1 |    |    |    |    |
   -----------------

Step 2: Fill in the K-map with the given minterms

The function F(x,y,z) is given in terms of minterms (the √ symbol is often used to denote the minterms of a function). We will place a 1 in the cells corresponding to the minterms listed: 0, 1, 2, 4, 5, 6, 8, 9, 12, 13, and 14.

Here is how the K-map looks after placing the 1s:

      yz
      00  01  11  10
   -----------------
x=0 | 1  | 1  |    | 1  |
   -----------------
x=1 | 1  | 1  | 1  | 1  |
   -----------------

Step 3: Group the 1s into the largest possible groups

We group the 1s into the largest possible groups of 1, 2, 4, or 8 to simplify the function. Each group must be a power of 2 and the groups must be rectangular in shape. Overlapping groups are allowed.

Here are the groups:

1. Group of four covering minterms 0, 1, 4, 5 (top left corner).
2. Group of four covering minterms 4, 5, 6, 7 (top right corner).
3. Group of four covering minterms 8, 9, 12, 13 (bottom left corner).
4. Group of two covering minterms 2, 6 (middle top row).

Step 4: Write the simplified Boolean function

Each group represents a term in the simplified function. We determine the variables that are constant within each group and write a term for that group.

1. For the group of four in the top left corner, x=0 and z=0 are constant, so the term is \bar{x}\bar{z}.
2. For the group of four in the top right corner, x=0 and y=1 are constant, so the term is \bar{x}y.
3. For the group of four in the bottom left corner, x=1 and z=0 are constant, so the term is xz.
4. For the group of two in the middle top row, y=0 and z=1 are constant, so the term is \bar{y}z.

Now, we combine these terms with the OR operation to get the simplified function:

F(x,y,z) = \bar{x}\bar{z} + \bar{x}y + xz + \bar{y}z

Step 5: Verify the simplified function

To ensure that we have simplified the function correctly, we can compare the original minterms with the minterms generated by the simplified function. If they match, our simplification is correct.

Summary Table

Here is a table summarizing the groups and the corresponding terms:

Group	Minterms Covered	Constant Variables	Simplified Term
1	0, 1, 4, 5	x=0, z=0	\bar{x}\bar{z}
2	4, 5, 6, 7	x=0, y=1	\bar{x}y
3	8, 9, 12, 13	x=1, z=0	xz
4	2, 6	y=0, z=1	\bar{y}z

The final simplified Boolean function is:

F(x,y,z) = \bar{x}\bar{z} + \bar{x}y + xz + \bar{y}z

This is the simplest form of the given Boolean function using the K-map method.