# Deriving full adder sum and carry outputs using boolean algebra

• dannyres
In summary, the student is struggling to write the sum and output of a full adder using XOR logical functions and Karnaugh maps. They have obtained the expression from the Karnaugh map but are having difficulty rearranging it into the expected form. They have made some progress but are stuck and seeking suggestions. They also ask for techniques on transitioning from traditional minterm implementations to XOR logic.

## Homework Statement

Hi, I am trying to write the sum and output of a full adder in terms of XOR logical functions using boolean logic and Karnaugh maps. I've got the expressions from the Karnaugh maps fine but I can't seem to rearrange them into the expected form shown at the end of my working.

Explained above.

## The Attempt at a Solution

It's going to be difficult writing my working here but hopfully it is clear:

My equation obtained from the Karnaugh map using a minimized SOP:
Sum = NOT(A).NOT(B).C + NOT(A).B.NOT(C) + A.B.C + A.NOT(B).NOT(C)
Sum = NOT(A).(NOT(B).C + B.NOT(C)) + A.(B.C + NOT(B).NOT(C))
Sum = NOT(A).(B XOR C) + A.(B.C + NOT(B).NOT(C))

..
...
...

Sum = NOT(NOT(A).B + A.NOT(B)).C + (NOT(A).B + A.NOT(B)).NOT(C)
Sum = (A XOR B) XOR C

This is where I'm trying to get. I've tried going both backwards and forwards but I just don't know where to go in-between. Any suggests to get me past the 3rd step I'm at?

Thank you,
Dan

I've made some significant progress but I've become stuck again.. here is what I've got:

#### Attachments

• sum.png
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Ouch. Welcome to the PF. This is a 3-bit full-adder? Could you post the truth table, including the carry bit? But then how do K-maps help in an XOR implementation of the truth table? I don't think I've ever had to morph into XOR logic (must be an academic thing?) -- what are the techniques for going from traditional minterm implementations (K-maps) to XOR logic?

## 1. How is a full adder designed using boolean algebra?

A full adder is designed using boolean algebra by using logic gates to manipulate boolean expressions. The boolean expressions are derived from the truth table of a full adder, and the logic gates are arranged in a way that represents the boolean expressions. This allows for the implementation of a full adder in hardware.

## 2. What are the boolean expressions for the sum and carry outputs of a full adder?

The boolean expression for the sum output of a full adder is A ⊕ B ⊕ C, where A and B are the two input bits and C is the carry input. The boolean expression for the carry output is (A ∧ B) ∨ (C ∧ (A ⊕ B)).

## 3. How can boolean algebra be used to simplify the expressions for the sum and carry outputs of a full adder?

Boolean algebra can be used to simplify the expressions for the sum and carry outputs by using the rules and laws of boolean algebra, such as the distributive law, De Morgan's laws, and the identity laws. By applying these rules, the expressions can be reduced to their simplest form.

## 4. Can a full adder be implemented using only boolean algebra without the use of logic gates?

No, a full adder cannot be implemented using only boolean algebra. While boolean algebra can be used to derive the expressions for the sum and carry outputs, it still requires the use of logic gates to physically implement the full adder in hardware.

## 5. What are the advantages of using boolean algebra to derive the sum and carry outputs of a full adder?

Using boolean algebra to derive the sum and carry outputs of a full adder allows for a systematic and logical approach to designing the full adder. It also allows for the simplification of the expressions, making them easier to understand and implement in hardware. Additionally, boolean algebra can be used to create more complex circuits by combining multiple full adders.