Homework Help: Logic gates - Electronics

1. Feb 14, 2012

Femme_physics

1. The problem statement, all variables and given/known data
Given the function:

http://img26.imageshack.us/img26/9718/elel0.jpg [Broken]

A) Write the truth table of the function F (A, B, C, D)

B) Present the function F (A, B, C, D) via Karnaugh Map

C) Express the function F as the sum of multiplications with minimum literals

D) Realize the minimized function F via logic gates

3. The attempt at a solution

I just wanna see if I got it right :)
http://img84.imageshack.us/img84/1940/elel1.jpg [Broken]

http://img577.imageshack.us/img577/1851/elel2.jpg [Broken]

Last edited by a moderator: May 5, 2017
2. Feb 14, 2012

Staff: Mentor

See your gate arrangement--you have used two NOT gates to twice produce B'. This is unnecessary duplication.

I can't say much about your Karnaugh map, I need to revise that topic myself. :( But I can see that your equation F= AB + B'C + B'C'D' does not match your truth table. Isn't it supposed to??

Last edited: Feb 14, 2012
3. Feb 14, 2012

Staff: Mentor

let's take as correct, your equation F= AB + B'C + B'C'D'

take out as a factor B' --> F = AB + B'(C + C'D')

consider that last term, B'(C + C'D')

when C is true, the bracketed term evaluates as true
when C is false, the bracketed term evaluates as D

so I think there should be some algebra reduction that allows you to make this

F = AB + B'(C + D')

4. Feb 14, 2012

Staff: Mentor

Here's how to go about demonstrating this. After you've been shown once, you'll probably be able to figure it out for yourself thereafter.

let's focus on the term in brackets,
C + ¬C¬D

EDITED:
Consider two basic logic relations:
you can OR anything with TRUE and it's still TRUE
you can AND anything with TRUE and it doesn't change its value
= C ( 1 + ¬D) + ¬C¬D

remove the brackets
= C + C¬D + ¬C¬D

take out a common factor ¬D
= C + ¬D (C + ¬C)

what's in brackets evaluates as always TRUE, so simplifies to
= C + ¬D

Last edited: Feb 14, 2012
5. Feb 14, 2012

Femme_physics

I never heard this "Or everything" rule in our Boolean algebra. Could our teacher only want us to use the basic list he gave us?

Point taken.

I don't know how to relate truth tables to functions, only functions to Karnaugh Maps and Karnaugh Maps to truth tables. Is it even possible?

Last edited: Feb 14, 2012
6. Feb 14, 2012

I like Serena

I believe that expression does match the truth table.

Btw, you can construct a truth table from a function.
If you want, introduce a couple of intermediary results.
Then calculate the result of the function for each combination of A, B, C, and D.

Which basic list did he give you?
It should include that (1 + a) is always true, that is, it is equal to 1.

Btw, can't you put in a drawing of something? Anything? A beetle would do.

Last edited: Feb 14, 2012
7. Feb 14, 2012

Staff: Mentor

Oops, oops, oops! I left out half the explanation in that step.

You can AND anything with TRUE and you don't change its value.

Sorry for the oversight.

Last edited: Feb 14, 2012
8. Feb 14, 2012

I like Serena

Uhh :uhh:... where's the typo?

9. Feb 14, 2012

Staff: Mentor

Fixed now.

10. Feb 15, 2012

Staff: Mentor

I overlooked the distinction between Ø and O so was just a little puzzled. Now I recognize you used Ø for your "don't care" states. So all is correct.

F = AB + B'(C + D')
⇔ F = AB + B'C + B'D'

11. Feb 16, 2012

Femme_physics

:) Thank you!

12. Feb 16, 2012

Ninty64

The simplification in the equation could have also been obtained by grouping the four corners on the Karnaugh map instead of just boxes 0 and 8.

I didn't see anyone else mention it, so I thought I would throw that out there