# Can a circuit that shows an error if a 4-bit number is higher than 9 be simplified to "(DC) + (DB)"?

I need to build a circuit that accepts a 4-bit number and turns on a LED (representing an error) if the given number is higher than 9.

As I understood, it should represent an "error display for a BCD counter". Since a BCD accepts a number of 4 bits (0-15) but it only counts from 0-9, I should turn on a LED that represents the presence of an error on the counter.

So, I followed the steps below:

1. Built the truth table;
2. Built the expression based on the truth table;
3. Built the Karnaugh Map;
4. Built the simplified expression from the Karnaugh Map;

1. Building the Truth Table

Note: the X column represents the output that shows an error (i.e. the LED)

2. Building the expression based on the Truth Table

So, based on the Truth Table, I got the following expressions:

Note: consider the (!X) as a NOT X

(D)(!C)(B)(!A) + (D)(!C)(B)(A) + (D)(C)(!B)(!A) + (D)(C)(!B)(A) + (D)(C)(B)(!A) + (D)(C)(B)(A)

3. Building the Karnaugh Map

So, as you guys can see, I found two quartets.

4. Building the simplified expression from the Karnaugh Map

As I found two quartets, this is the final expression I found:

(DC) + (DB)

The conclusion and the final question

So, after all these steps, I tried to apply the Truth Table values into my simplified expression (ignoring the A column, as the simplified expression doesn't contain it). It seems to be correct after all.

My question is:

Let's suppose someone get close to me and says:

"I've built a BCD counter. It should only count from 0 to 9. If a number higher than 9 is given, then there is an error. I need YOU to build ANOTHER circuit. I'll give you the 4-bit number that entered on my circuit and then you need to show an error in case this number is higher than 9."

Considering my final expression, (DC) + (DB), should I just ignore the port A, since there is no A in my final simplified expression? Does it mean I can just leave A without being connected to any wire?

This is correct.

(DC) + (DB) = D(C + B)

• ignore the fault, (input > 9) . x= don't care? or are there consequences?
• display a fault ( error detect ) is it useful?
• blank the display with flashing Dot or something( from erroneous data )
• use a HEX LED converter and show result as a,b,c,d,e,f ( is that useful? )

Only you can tell if there are consequences of a fault condition but often fault detection is a good thing if you do something with it. Sometimes this is only done on a power On reset with JTAG tests. The higher the % fault detection (metric for Test Engineers, the better the design.

• Thank you, Sunny! So from what I understood, what I should do really depends on what I'm building, right? Of course supposing this is gonna be implemented by a physical and real circuit. Since I'm a Computer Science student, this exercise is just a theoretical one. So, in this case, only turning on a LED saying "hey, you got an error" seems to be enough. Apr 12, 2019 at 17:37
• FWIW , yes....... in real applications they use non BCD for other purposes like higher level system error codes Apr 12, 2019 at 17:38

As the A input does not matter it is not necessary to decode it.

• Unless A is your boss saying sometimes you need to display when there is a faulty data. Yesm this may be insignficant in the overall project Apr 12, 2019 at 16:23