# Switching Algebra proof question

• ilikescience
In summary, to prove the equality X Y' + Y Z' + X' Z = X' Y + Y' Z + X Z', you only need to use the switching axioms and theorems listed on the second slide of the given link. By using distribution and the identity aa' = 0, you can easily show that the two sides are equal without needing DeMorgan or other manipulations.
ilikescience

## Homework Statement

Prove with algebraic manipulation the following equality: X Y' + Y Z' + X' Z = X' Y + Y' Z + X Z'

## Homework Equations

All you need to know to prove it are the switching axioms and theorems listed on the second slide of http://meseec.ce.rit.edu/eecc341-winter2001/341-12-13-2001.pdf" .

## The Attempt at a Solution

xy' + yz' + x'z = (xy' + yz' + x'z)' ' involution
= ((x'+y)(y'+z)(x+z'))' DeMorgan
= (xyz + x'y'z')' distribute and use aa' = 0
= (x' + y' + z')(x + y + z) DeMorgan
= (xy' + yz' + x'z) + (x'y + y'z + xz') distribute and use aa' = 0

since we have a = a + b then b = a or b = 0 by idempotency or identities, but I think we can show b ~= 0 if x, y ,or z are not all 0 or 1.

I feel like there might be an easier way. What do you think?

Last edited by a moderator:
all you need are distribution ,aa=a and a+a'=1

there is no need for DeMorgan or anything else

## 1. What is Switching Algebra?

Switching Algebra is a mathematical system used to analyze and design digital circuits. It uses a set of rules and symbols to represent and manipulate logical expressions, making it useful for solving problems related to digital logic.

## 2. What is a proof in Switching Algebra?

A proof in Switching Algebra is a series of logical steps that demonstrate the validity of a statement or equation. It involves applying the rules of the algebra to manipulate the given expressions and reach a logical conclusion.

## 3. How is a Switching Algebra proof different from a traditional proof?

A Switching Algebra proof is different from a traditional proof in that it uses logical symbols and rules specific to the algebra, rather than traditional mathematical operations. It also focuses on analyzing and manipulating logical expressions, rather than numerical values.

## 4. What are some common rules used in Switching Algebra proofs?

Some common rules used in Switching Algebra proofs include De Morgan's Laws, the Distributive Law, and the Idempotent Law. These rules allow for the simplification and manipulation of logical expressions, making it easier to reach a logical conclusion.

## 5. How is Switching Algebra used in practical applications?

Switching Algebra is used in practical applications such as circuit design and analysis, digital signal processing, and computer programming. It allows for the efficient and systematic manipulation of logical expressions, making it essential for the development of digital technology.

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