Implementing boolean functions with decoder and external gate

[CoolDude420]

Homework Statement

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

Homework Equations

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?

 

[CoolDude420]

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?

 

[CoolDude420]

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?

 

[Tom.G]

...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.

 

[CoolDude420]

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]

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]

The missing variable is a 'Don't Care.'

...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.