18 votes

Which are the essential prime implicants of the following Boolean function?

$f(a, b, c)= a' c+ ac'+b' c$

- $a' c$ and $ac'$
- $a' c$ and $b' c$
- $a' c$ only.
- $ac'$ and $bc'$

7 votes

Best answer

$f(a,b,c) = a'c+ac'+b'c$

We can write these product of sum terms into canonical product of sum form.

$f(a,b,c) = \underbrace{a'b'c}_{001}+\underbrace{a'bc}_{011}+\underbrace{ab'c'}_{100}+\underbrace{abc'}_{110}+\underbrace{ab'c}_{101}+\underbrace{a'b'c}_{001}$

$f(a,b,c) = \sum(1,3,4,5,6)$

Now, we can draw the k-map for these minterms.

- Prime implicant of $f$ is an implicant that is minimal - that is, the removal of any literal from product term results in a non-implicant for $ f$.
- Essential prime implicant is an prime implicant that cover an output of the function that no combination of other prime implicants is able to cover.

Prime implicants are$:a'c,b'c,ab',ac'$

Essential prime implicants are$:a'c,ac'\:\text{(green color)}$.

References:

0

@Lakshman Patel RJIT @techbd123

**No. of Implicants = No. of Minterms = 5**

**Please, correct me if i am wrong.**

1

Number of implicants $=$ number of minterms $+$ number of subcubes of size $2$

Number of implicants $= 5 + 4 = 9$

PS: Number of implicants $=$ number of subcubes of size $2^{0} \:+$ number of subcubes of size $2^{1} +$ number of subcubes of size $2^{2} +$ number of subcubes of size $2^{3}$

0

If prime implicant is the group of one's, then what is the difference between prime implicant & implicant

0

From the above comment

$\implies$ All prime implicant is also implicant but the reverse is not true.

19 votes

3

Just to see it visually ;)

Clearly $A\bar{C}$ and $\bar{A}C$ cover cells that are not covered by any others.

0

Essential prime implicant: A prime implicant that includes one or more distinguished one cells. Essential prime implicants are important because a minimal sum contains all essential prime implicants.

Reference: