Step 2: prime implicant chart[edit]
None of the terms can be combined any further than this, so at this point we construct an essential prime implicant table. Along the side goes the prime implicants that have just been generated, and along the top go the minterms specified earlier. The don't care terms are not placed on top—they are omitted from this section because they are not necessary inputs.
To find the essential prime implicants, we run along the top row. We have to look for columns with only 1 "X". If a column has only 1 "X", this means that the minterm can only be covered by 1 prime implicant. This prime implicant is essential.
4 8 10 11 12 15 ⇒ A B C D m(4,12)* 
⇒ — 1 0 0 m(8,9,10,11) 
⇒ 1 0 — — m(8,10,12,14) 
⇒ 1 — — 0 m(10,11,14,15)* 
⇒ 1 — 1 —
For example: in the first column, with minterm 4, there is only 1 "X". This means that m(4,12) is essential. So we place a star next to it. Minterm 15 also has only 1 "X", so m(10,11,14,15) is also essential. Now all columns with 1 "X" are covered.
The second prime implicant can be 'covered' by the third and fourth, and the third prime implicant can be 'covered' by the second and first, and neither is thus essential. If a prime implicant is essential then, as would be expected, it is necessary to include it in the minimized boolean equation. In some cases, the essential prime implicants do not cover all minterms, in which case additional procedures for chart reduction can be employed. The simplest "additional procedure" is trial and error, but a more systematic way is Petrick's method. In the current example, the essential prime implicants do not handle all of the minterms, so, in this case, one can combine the essential implicants with one of the two non-essential ones to yield one equation:
{\displaystyle f_{A,B,C,D}=BC'D'+AB'+AC\ }[11]![f_{A,B,C,D} = BC'D' + AB' + AC \](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbda4845860dafa04f77d705abb843ada7e8f30e "f_{A,B,C,D} = BC'D' + AB' + AC \")
or
{\displaystyle f_{A,B,C,D}=BC'D'+AD'+AC\ }
Both of those final equations are functionally equivalent to the original, verbose equation:
{\displaystyle f_{A,B,C,D}=A'BC'D'+AB'C'D'+AB'C'D+AB'CD'+AB'CD+ABC'D'+ABCD'+ABCD.\ }![f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD. \](https://wikimedia.org/api/rest_v1/media/math/render/svg/173851685446c00c7e1eba8069f19f9359db94fd "f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD. \")
The Quine McCluskey (QM) Method is a logical technique used to simplify Boolean algebra expressions. It is named after its developers, Willard Van Orman Quine and Edward J. McCluskey.
The QM Method uses a systematic approach to identify and eliminate redundant terms in a Boolean expression. It starts by creating a truth table and then uses a process of combining terms and eliminating duplicates to create a simplified expression.
The QM Method offers several advantages, including the ability to handle large and complex Boolean expressions, the guarantee of finding the most simplified expression, and the potential to reduce the number of terms and variables in the expression.
The QM Method is commonly used in digital electronics, specifically in the design and optimization of logic circuits. It is also used in software engineering for simplifying Boolean expressions in programming languages.
While the QM Method is a powerful tool for simplifying Boolean expressions, it does have some limitations. It can only be used for expressions with a limited number of variables, and it may not always produce the most efficient result. Additionally, the process can be time-consuming for complex expressions.