**Implicants**

Let an SOP function be expressed in the form

*f*(

*x*

_{1},

*x*

_{2},…,

*x*

_{m}) =

*x*

_{1}

*x*

_{2}….

*x*

_{n}+

*x*

_{1}

*x*

_{2}….

*x*

_{m}(2.27)

Let, at least, any one of the
product terms, say,

*x*_{1},*x*_{2},*x*_{3}, …,*x*_{m}= 1. Then, we observe that the given function*f*(*x*_{1},*x*_{2},*x*_{3}, …,*x*_{m}) = 1. In such a case, we call the product term*x*_{1},*x*_{2},*x*_{3}, …,*x*_{m}*as an**implicant*of the given logic function*f*(*x*_{1},*x*_{2},*x*_{3}, …,*x*_{m}).
We now define the term implicant. Let

*f*(*x*_{1},*x*_{2},*x*_{3}, …,*x*_{m}) be an*m*-variable SOP function and*S*be the product term*x*_{1}*x*_{2}….*x*_{n }in that function*.*We now state that*S is an implicant of the function f if and only if the value S is equal to**1**so that the value of the function f is also equal to**1*. It can be seen that this definition is quite lengthy. To explain this, consider the following example. Let*f*(

*x*

_{1},

*x*

_{2},

*x*

_{3},

_{ }

*x*

_{4}) =

*x*

_{1}′

*x*

_{2}

*x*

_{3}

*x*

_{4}+

*x*

_{1}

*x*

_{2}

*x*

_{3}

*x*

_{4}+

*x*

_{1}

*x*

_{2}

*x*

_{4}+

*x*

_{1}

*x*

_{3}

*x*

_{4 }(2.28)

In Eq. (2.28), we have four product terms on the RHS. Let any one or more of the above four terms give an output of 1. Then, this implies that

*f*= 1. For example, if

*x*

_{1}

*x*

_{2}

*x*

_{4}

*= 1, then*

*f*= 1. Similarly, when

*x*

_{1}′

*x*

_{2}

*x*

_{3}

*x*

_{4}

*= 1 or*

*x*

_{1}

*x*

_{2}

*x*

_{3}

*x*

_{4}

*=*1, then also

*f*= 1, and so on. Thus each one of the above terms is an implicant. Notice also that

*a minterm will be an implicant, but an implicant need not be a minterm.*For example, the terms

*x*

_{1}

*x*

_{2}

*x*

_{3}

*x*

_{4}and

*x*

_{1}′

*x*

_{2}

*x*

_{3}

*x*

_{4}

*are implicants as well as minterms. But, the implicants*

*x*

_{1}

*x*

_{3}

*x*

_{4}and

*x*

_{1}

*x*

_{2}

*x*

_{4}are not minterms.

If an implicant must make a function equal
to 1, then a term that makes a function equal to 0 is not an implicant of that function. Consider the function

*f*(

*a, b, c, d*)

*=*

_{ }

*abd + abc*+

*acd*(2.29)

In this function, whenever

*b*=*d*= 0,*f*= 0. This means that the term*b'd'*is not an implicant of*f*.**Prime Implicants**

*A prime implicant is an implicant from which if we delete any variable (or literal), then it can no longer be considered as an implicant*. For example, consider the term

*abd*in Eq. (2.28). From this equation, if we remove any one of the terms (i.e.,

*a*,

*b*, or

*d*), then the resulting product term will no longer imply

*f.*That is, if

*a*is removed, then the resulting term

*bd*is no longer an implicant of the function

*f*. Then, we say that the term

*abd*is a prime implicant of

*f*. Similarly, the other terms (i.e.,

*abc*and

*acd*) are also prime implicants of

*f*.

**Essential Prime Implicant**

*A prime implicant is said to be essential, if a minterm in an SOP expression is covered by only one prime implicant.*For example, let us consider the K-map shown in Fig. 2.25. We find that minterm

*m*

_{2}is covered by prime implicant

*A*only. So, we call

*A*as an essential prime implicant. Similarly, minterm

*m*

_{12}is covered only by prime implicant

*B*, and hence

*B*is an essential prime implicant. We also find that minterms

*m*

_{5}and

*m*

_{15}are not covered by any other prime implicants; hence

*C*is also an essential prime implicant. Summarizing the discussions, we may now state that:

- Any
minterm or combinations of minterms, which make an SOP function equal to
**1**is an implicant. - If,
from an implicant, a literal removed makes it
*not*to be - In a reduced function, if the removal of a prime implicant changes the structure of the function, then that prime implicant is essential in forming the function.

Figure 2.26 highlights the implicants as

*A*,

*B*,

*C*,

*D*, and

*E*, respectively. The corresponding minterms are

*a*'

*cd*,

*abd*,

*ab’c*,

*bc*'

*d*, and

*acd*, respectively. Of the five implicants, all are prime implicants. However, implicants

*D*, and

*E*are not essential, as the minterms in them are already covered by

*A*,

*B*, and

*C*. These prime implicants are called

*redundant*terms.