# Implementing boolean functions with decoder and external gate

• CoolDude420
In summary: In this case, Z is set to False and the equation for F1 is simplified to X Z.F1 can be satisified if either X or Z are True, but not both.F1 can be satisified if either X or Z are True, but not both.
CoolDude420

## Homework Statement

Design an combinational circuit using a decoder and external gates defined by the boolean functions F1, F2, F3(see picture)

## The Attempt at a Solution

I'm quite confused as to the exact method in doing this. I understand that a decoder takes n inputs and produces 2^n outputs. The combination of the n inputs correspond to binary numbers, whatever binary numbers the inputs make, the corresponding output line will be 1. For example if x=0, y=0, z=0, that repersents binary 0, so D0 will be 1.

My confusion arises when there are minterms in F1,F2,F3 that only have 2 variables. Do I just set the third, non present variable to 0? Any ideas?

I went ahead and implemented F1 to my understanding.

I got F1 = Σm(0,5).
The answers given to us is F1 = Σ(0,5,7)
Any ideas

Also, is it better to write Σm or is just Σ fine?

I was always told that with decoders that each output equation contains all of the input variables but since the one of the minterms in the F1 output equation doesn't contain all variables, how do I deal with that?

CoolDude420 said:
...one of the minterms in the F1 output equation doesn't contain all variables, how do I deal with that?

The missing variable is a 'Don't Care.' That means the result of the minterm is the same regardless of the value of the missing variable.

Tom.G said:
The missing variable is a 'Don't Care.' That means the result of the minterm is the same regardless of the value of the missing variable.

I ended up connecting D0, D5 and D7 to an OR gate which basically resulted in the y term being canceled out by simplification.

CoolDude420 said:
I was always told that with decoders that each output equation contains all of the input variables but since the one of the minterms in the F1 output equation doesn't contain all variables, how do I deal with that?
Also how did you figure out that it's a don't care term without simplification

Tom.G said:
The missing variable is a 'Don't Care.'

CoolDude420 said:
...how did you figure out that it's a don't care term without simplification

F1 = ## \bar X \bar Y \bar Z##
or
##X Z##​

The equation for F1 says there are two different conditions, either of which can satisify it:
• X, Y, Z are all False
-or-
• X and Z are both True
Restating it; For F1 to be satisified, it is sufficient that both X and Z are True, nothing else matters and there is no reason to include anything else in the minterm; all other variables are extraneous.

By the same token, F1 can also be satisified if X, Y, Z are all False.

## 1. How can boolean functions be implemented with a decoder and external gate?

Boolean functions can be implemented with a decoder and external gate by using the decoder to generate all possible combinations of input values and then using the external gate to select the desired output based on the boolean function being implemented.

## 2. What is the advantage of using a decoder to implement boolean functions?

The advantage of using a decoder is that it simplifies the implementation process by automatically generating all possible combinations of input values, which can save time and reduce the potential for errors.

## 3. Can any boolean function be implemented using a decoder and external gate?

Yes, any boolean function can be implemented using a decoder and external gate as long as the function can be expressed in terms of AND, OR, and NOT operations.

## 4. What is the role of the external gate in implementing boolean functions?

The external gate's role is to select the desired output based on the boolean function being implemented. It takes in the output values from the decoder and uses them to determine which output should be selected.

## 5. Are there any limitations to implementing boolean functions with a decoder and external gate?

The main limitation of using a decoder and external gate is that it requires additional hardware components, which can increase the complexity and cost of the implementation. Additionally, this method may not be efficient for implementing more complex boolean functions with a large number of input variables.

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