How Do You Derive the Sum Formula for a Full Adder?

[King]

Hi,
I've been trying to work out the formula for the sum for the full adder logic, however have come across a gap which I don't know how to fill.
S = (¬A.¬B.C) + (¬A.B.¬C) + (A.¬B.¬C) + (A.B.C)
S = ¬A.(¬B.C + B.¬C) + A.(¬B.¬C)
S = ¬A.(B $$\oplus$$ C) + A.( do not know what to do at this point to reach the next stage
S = (A $$\oplus$$ B) $$\oplus$$ C

Does anyone know how to get to that last statement?

Thanks for any help.

 

[berkeman]

Hi,
I've been trying to work out the formula for the sum for the full adder logic, however have come across a gap which I don't know how to fill.
S = (¬A.¬B.C) + (¬A.B.¬C) + (A.¬B.¬C) + (A.B.C)
S = ¬A.(¬B.C + B.¬C) + A.(¬B.¬C)
S = ¬A.(B $$\oplus$$ C) + A.( do not know what to do at this point to reach the next stage
S = (A $$\oplus$$ B) $$\oplus$$ C

Does anyone know how to get to that last statement?

Thanks for any help.

It's probably easiest to draw a Karnaugh map for the function, and use that to help your intuition in the reduction. Have you drawn a K-map for this yet?

 

[King]

Nope. I have never used one before.

Edit: How come this received a warning? This isn't homework or coursework. It's just studying.

 

[berkeman]

Nope. I have never used one before.

Edit: How come this received a warning? This isn't homework or coursework. It's just studying.

Studying is coursework. Even if it's for self-study, it belongs in the Homework Help forums, and needs to follow the rules here. You can check out the Rules link at the top of the page for clarification.

The K-map is a great tool for helping you visualize logic reduction. Here's an intro to get you started -- there are lots of other learning resources for K-maps on the web:

http://en.wikipedia.org/wiki/Karnaugh_map

.

 

[King]

Studying is coursework. Even if it's for self-study, it belongs in the Homework Help forums, and needs to follow the rules here. You can check out the Rules link at the top of the page for clarification.

That doesn't make sense since every thread in these forums is about help to understand something, so one could always link it to a course since there are courses for everything, and thus everything should therefore be in the Homework Help forums. But anyway...

Does anyone know how to solve this without the K-map since I don't really have colouring pencils to create this diagram, just a pen? Also, how would one solve this algebraically?

 

[Defennder]

Why do you need coloured pencils for the Karnaugh map?