Help with a Linear Algebra problem please

[MyoPhilosopher]

Homework Statement

Testing Linear Algebra Rules on following statement

Relevant Equations

R1. u + v = v+u
R2. u + (v + z) = (u + v) + z
R3.1 z + v = v
R3.2 u + z = v

For the following statement:
V = R ≥ 1; x ⊕ y = max (x,y), with z = 1

My attempt is as follows:

Should R3 be z ⊕ (x ⊕ y)?
I am confused at to the notation of this rule. Moreover, I am struggling to find examples and answers of such problems in linear algebra online.
Should I always view such questions (with x,y) as x representing "u" and y representing "v"?

Thank you

 

[PeroK]

I'm guessing to some extent about what you are doing.

The first problem is you seem to be using $z$ as an arbitrary real number and as a "zero"?

The other problem is that sometimes you use $x + y = x \oplus y = max(x, y)$ and sometimes you use normal addition. For example, what I believe you are trying to test is whether, for all $x, y, z$:

$(x \oplus y) \oplus z = x \oplus (y \oplus z)$

 

[MyoPhilosopher]

I'm guessing to some extent about what you are doing.

The first problem is you seem to be using $z$ as an arbitrary real number and as a "zero"?

The other problem is that sometimes you use $x + y = x \oplus y = max(x, y)$ and sometimes you use normal addition. For example, what I believe you are trying to test is whether, for all $x, y, z$:

$(x \oplus y) \oplus z = x \oplus (y \oplus z)$

Yes you are correct I severely overlooked that one...
Would you mind explaining to me what the 3rd rule attempts to prove mathematically:
Should it be z ⊕ (x ⊕ y) = x ⊕ y
or z ⊕ x = x AND THEN z ⊕ y = y

 

[PeroK]

Yes you are correct I severely overlooked that one...
Would you mind explaining to me what the 3rd rule attempts to prove mathematically:
Should it be z ⊕ (x ⊕ y) = x ⊕ y
or z ⊕ x = x AND THEN z ⊕ y = y

I assume you want to use $z$ for the additive identity (normally denoted $0$) and in this case show whether or not the number $1$ has the properties of an additive identity. Let's use $z$ for this, although I've never seen that notation before. In case, let's use $u, v, w$ as our vectors. You need to show that for all $u$ we have:

$u \oplus 1 = u$
$1 \oplus u = u$