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Let $f(x, y, z)=\bar{x} + \bar{y}x + xz$ be a switching function. Which one of the following is valid?

  1. $\bar{y} x$ is a prime implicant of $f$

  2. $xz$ is a minterm of $f$

  3. $xz$ is an implicant of $f$

  4. $y$ is a prime implicant of $f$

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44 votes
44 votes

In sum of terms, any term is an implicant because it implies the function. So, $xz$ is an implicant and hence C is the answer. Still, lets see the other options. 

If no minimization is possible for an implicant (by removing any variable) it becomes a prime implicant. 

If a prime implicant is present in any possible expression for a function, it is called an essential prime implicant. (For example in K-map we might be able to choose among several prime implicants but for essential prime implicants there won't be a choice). 

So, $f = x' + y'x + xz$
$= y' + x' + z$ (could be also derived using algebraic rules as in http://www.ee.surrey.ac.uk/Projects/Labview/boolalgebra/ )

So, the prime implicants are $x', y'$ and $z$. Being single variable ones and with no common variables, all must be essential also.  

Choice a) False - $y'$ is a prime implicant and hence, $y'x$ is just an implicant but not prime. 
Choice b) False - $xz$ is not a minterm. A minterm must include all variables. So, $xyz$ is a minterm so, is $xy'z$, but not $xz$. 
Choice d) False - $y'$ is a prime implicant not $y$. 
 

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33 votes
33 votes

Answer: C

f(x,y,z) = x' + y'x + xz

An implicant of a function is a product term that is included in the function.

so x', y'x and xz ,all are implicants of given function.

A prime implicant of a function is an implicant that is not included in any other implicant of the function. 

option a)   y'x is not a prime implicant as it is included in xz [ xy'z+ xyz]

option d) y is not a prime implicant as it include in both x' and xz.

a product term in which all the variables appear is called a minterm of the function

option b) xz is not a minterm

6 votes
6 votes

$xz$ is an implicant and $\neg y$ is both prime and essential prime implicant. The sop would be $z+\neg x+\neg y$. 

Implicant: Something that implies a function is its implicant
Prime implicant: The most reduced (minimal) implicant
Essential prime implicant: The prime implicant which cannot be avoided in any SOP

Ref: https://en.wikipedia.org/wiki/Implicant

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