# Introduction

In the previous lectures, you have learnt the Boolean algebra and K-Map methods of solving Boolean expressions.

These methods are simple and easy for less number of variables say upto four. But for more than 4 variable equations, these become tedious. So, another method known as Quine Mc-Clusky or Tabulation Method is used for solving Boolean expression involving four or more variables.

## Quine Mc-Clusky or Tabulation Method

This is again a simple method but requires lot of concentration while looking for minterms with 1-bit change. A little distraction of mind will lead to committing a mistake.

### Various steps involved in simplification are:

- Preparing index value based on number of 1’s in the minterms of the equation including don’t care terms.
- Rearranging the minterms as per index value
- Checking minterms from one group to another for a 1-bit change and ticking such terms.
- Repeating step-3 till such time no further pairing of minterms are possible. All such terms are called as
**prime implicants**terms - Preparing PI chart to find
**essential prime implicants**

**Example:**

**Simplify Y(A,B,C,D) = ∑m(0,2,3,6,7,8,10,12,13) using McClusky or the Tabulation method**

**Solution**

**Step-1:** Prepare index value (i.e. find number of 1’s in each minterm)

minterm | mo | M2 | M3 | M6 | M7 | M8 | M10 | M12 | M13 |

Binary | 0000 | 0010 | 0011 | 0110 | 0111 | 1000 | 1010 | 1100 | 1101 |

Index | 0 | 1 | 2 | 2 | 3 | 1 | 2 | 2 | 3 |

**Step-2:** Group the minterms as per index value

At this point, one can start combining minterms with other minterms. If two terms vary by only a single digit changing, that digit can be replaced with a dash indicating that the digit doesn’t matter. Terms that can’t be combined any more are marked with an asterisk (*)

**Step-3:** Preparing PI Char for finding Essential Prime Implicants

So, the final solution of the expression Y(A,B,C,D) = ∑m(0,2,3,6,7,8,10,12,13) is:

Y = A’C + B’D’ + ABC’

Exercise:

Solve the following Boolean expressions using Tabulation method

- Y(A,B,C,D) = ∑m(0,1,2,5,6,7,8,9,10,14)
- Y(A,B,C,D) = ∑m(0,5,8,9,10,11,14, 15)
- Y(A,B,C,D) = ∑m(0,1,3,7,8,9,11,15)
- Y(A,B,C,D) = ∑m(4,6,9,10,11,13) + ∑d(2,12,15)