The Boolean Algebra XOR Problem: Are These Expressions Equivalent?

[Rectifier]

The problem
This is not a complete homework problem. I am at the last step of the solution to a long problem and only interested to know whether these following expressions are equivalent.

My answer:
$a \oplus ab \oplus ac$

Answer in my book:
$a \oplus b \oplus c$

The attempt
I have tried rewriting that expression and get
$a \oplus ab \oplus ac = a \oplus a(b \oplus c)$

I am not advancing any longer from there. I am surely missing some algebraic law for boolean expressions. Could you please help me?

 

[cpscdave]

Did you try building a truth table?
If you do you'll see that your answer is different from the book's answer.
For example ABC = 0,1,0
The book answer gives 1
Your answer results in 0

 

[jedishrfu]

Part of your problem may be your understanding of operator precedence.

It seems to vary slightly across programming languages. For C, Java and Javascript it is:

NOT > AND > XOR > OR

a XOR ( a AND b) XOR ( a AND c)

with XOR equivalent to:

x XOR y = (x OR y) AND (NOT ( x AND y) // mentor note: fixed the expression (thx cpscdave)

reworking the second and third terms should get you to the goal.

 

[Rectifier]

Well, I guess I am doing something wrong then. :/ Thanks for the help.

 

[cpscdave]

x XOR y = (x OR y) AND (NOT ( x OR y)

Think you mean
(x OR y) AND [NOT ( x AND y)]

or alternatively

(x OR y) AND ( NOT x or NOT y)

 

[jedishrfu]

Yes you are right my apologies.

I got my up and down arrows confused in my conversion to ands and ors.

I fixed my post.

 

[SammyS]

a ⊕ b ⊕ c does have a rather interesting truth table.

 

[NascentOxygen]

My answer:
$a \oplus ab \oplus ac$

Answer in my book:
$a \oplus b \oplus c$

The attempt
I have tried rewriting that expression and get
$a \oplus ab \oplus ac = a \oplus a(b \oplus c)$

I am not advancing any longer from there. I am surely missing some algebraic law for boolean expressions. Could you please help me?

I would proceed by converting one XOR at a time to its equivalent in AND and OR, starting with this:

$\mathrm {a \oplus ab \oplus ac = a (\overline {ab \oplus ac }) + \overline a (ab \oplus ac)}$