Q.Simplify the Boolean expression Y= $\Sigma m(0,2,6,8,10,12,14)+\Sigma d(4,9,13)$ using K-map. Draw the NAND gate equivalent circuit to realize the simplified equation.

A Karnaugh map (K-map) is a graphical tool for simplifying Boolean expressions quickly and reliably. It is a grid of cells, one cell for every possible input combination — a 3-variable map has 8 cells and a 4-variable map has 16 cells. The rows and columns are labelled in Gray-code order so that any two horizontally or vertically adjacent cells differ in only one variable.

Simplification is done by looping — encircling adjacent 1s in groups whose size is a power of two. A pair of two adjacent 1s eliminates one variable, a quad of four eliminates two variables, and an octet of eight eliminates three variables. Groups may overlap, and because the map wraps around, cells on opposite edges or the four corners are treated as adjacent, a technique called rolling. Larger groups give simpler terms, so the aim is to cover every 1 with the fewest, largest possible groups. …