2.4k views

A circuit outputs a digit in the form of $4$ bits. $0$ is represented by $0000$, $1$ by $0001$, …, $9$ by $1001$. A combinational circuit is to be designed which takes these $4$ bits as input and outputs $1$ if the digit $\geq$ $5$, and $0$ otherwise. If only AND, OR and NOT gates may be used, what is the minimum number of gates required?

1. $2$
2. $3$
3. $4$
4. $5$

edited | 2.4k views
+7

Answer would be (B) "3 Gate will be needed to design the o/p see below 0
Why you consider 3 input OR gate?
+1
@prateek

Your solution is wrong. We cannot take 3 IP OR as by default we have to take only 2 IP ones. Also terms beyond 9 will be don't cares.
0
Prateek's answer is right, Since min no. of gates have been asked so we can assume 3-i/p gate is available and for the don't care part , he has taken 1s in place of don't care.

or A + BC + BD => A + B(C + D) only 2 i/p gates are required.
0
A 4-bit number $abcd$ $\geq 5$ iff $a$ is 1 and at least one of the remaining bits $b, c, d$ is 1.

In other words, $a$ is 1 and $b + c + d$ is 1 where "$+$" is  OR. "And" becomes "$.$"

$a.(b + c + d) = 1$

You need 2 OR gates and 1 AND gate for this. So total = 3

Answer should be (B). As according to question, truth table will be like:

$A B C D$    $f$

$0 0 0 0$      $0$

$0 0 0 1$      $0$

$0 0 1 0$      $0$

$0 0 1 1$      $0$

$0 1 0 0$      $0$

$0 1 0 1$      $1$

$0 1 1 0$      $1$

$0 1 1 1$      $1$

$1 0 0 0$      1

$1 0 0 1$      1

$1 0 1 0$     do not care

$1 0 1 1$    do not care

$1 1 0 0$    do not care

$1 1 0 1$    do not care

$1 1 1 0$    do not care

$1 1 1 1$    do not care

Using this truth table we get  $3$ sub cube which are combined with following minterms  $A (8,9,10,11,12,13,14,15)$, $BD( 5,13,7,15)$ and $BC(6,7,14,15)$

So, $f = A+ BD +BC= A+ B(C+D)$

So, minimum gate required $2$ OR gate and $1$ AND gate $= 3$ minimum gate..

by Boss (17k points)
edited
+1
why we have consider 10 - 15 as don't care ?
+2
Thanks now i got it
+1
Why we have considered 10-15 nos. as don't cares ??
+1
why 10 to 15 are dont care?
+1
see the first line of question
0
In first line it's not clear that they are taking only upto 9!! then why don't care??
0

flash12 Don't care (X) refers to the set of input for which the output of the function does not matter or a set of inputs that is known to never occur. Now, that being said, technically it leaves a function non-deterministic as we do not know what output the function will give on these set of inputs. However, when a circuit is realized it can only be deterministic; meaning you're in a way forced to define the output for the function even at those positions for which the input is known not to occur.

Now, we do something clever, in order to minimize our circuitry we assume the values of these don't cares in such a way that the overall expression for the function has minimum literals and minterms and thus a smaller circuit.

Coming back to the question, if you closely look, you'll see that the octet formed for deriving term A in the answers just takes all those don't cares to be 1. This does not mean you have changed the function. The function is known to never take those values and we can thus play with them for our benefit. Just draw K-Map and you'll see.

Please note, this a very intuitive yet useful way of thinking about don't cares. I suggest that you read more about them from standard text

+4

10-15 is taken as don't care because the question says that if the "digit" >=5. From 10 to 15 they are not a digit number. So we don't care about them.

0
@MiNiPanda

Nothing is mentioned then why cant we take

$3-input$   $OR$  gate ?

Or is it like, nothing is mentioned hence we cant take $3-input$   $OR$  gate by ourselves ?
0

I am not much sure but I think by default fan in of gates should be taken as 2.

0
@MiNiPanda

Sir used it, just encountered in one of the vid. But comments above saying it wrong.
We should get output 1 for values>=5

Making truth table for problem
A     B     C     D     Op
0     0     0     0     0
0     0     0     1     0
0     0     1     0     0
0     0     1     1     0
0     1     0     0     0
0     1     0     1     1
0     1     1     0     1
0     1     1     1     1
1     0     0     0     1
1     0     0     1     1
1     0     1     0     X
1     0     1     1     X
1     1     0     0     X
1     1     0     1     X
1     1     1     0     X
1     1     1     1     X

Putting this in kmap and solving

49

Here crucial point is that we need to make pair of 8 elements using don’t cares also…so final expression is

A+BD+BC

A+B(C+D)

Hence we’ll use two OR gate and one AND gate so total 3 gates.

Ans (B) part.
by Active (4.7k points)

We should get output 1 for values>=5

Making truth table for problem

 A B C D Op 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 X 1 0 1 1 X 1 1 0 0 X 1 1 0 1 X 1 1 1 0 X 1 1 1 1 X

Putting this in kmap and solving Here crucial point is that we need to make pair of 8 elements using don’t cares also…so final expression is

A+BD+BC

• A+B(C+D)

Hence we’ll use two OR gate and one AND gate so total 3 gates.

Ans (B) part.

by Active (4.7k points)
reshown by

using K map simplified SOP AC' + B

values 10-15 are treated as dont care
by Loyal (8.7k points)
+1
Please show the k-map for this solution
+3
How are you getting AC' + B ? A+ BD +BC is correct !

1