I need my work checked(properties of operations)

Question

Let $x * y = |x + y|.$

$x * y = |x + y| = |y + x| = y * x,$ so $*$ is commutative.

$(x * y) * z = ||x + y| + z| = |x + |x + z|| = x * (y * z),$ so $*$ associative.

$x * e = |x + e| = x,$ so $e = 0.$ Further, $e * x = |0 + x| = x.$ So, $*$ has an identity.

$x * x' = |x + x'| = e,$ so $x' = e - x.$ Further, $x' * x = |e - x + x| = e.$ So, every element has an inverse.

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Let $x * y = |xy|.$

$x * y = |xy| = |yx| = y* x,$ so $*$ commutative.

$(x * y) * z = |(xy)z| = |x(yz)| = x * (y * z),$ so $*$ associative.

$x * e = |xe| = x,$ so, $e = 1$. Further, $e * x = |1x| = x$. So, $*$ has no identity.

$x * x' = |xx'| = e, $ so $x' = \frac ex$. Further, $x' * x = |\frac ex x| = e$. So, every element has an inverse.

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Let $x * y = \sqrt{x^2 + y^2}.$

$x * y = \sqrt{x^2 + y^2} = \sqrt{y^2 + x^2} = y*x$, so $*$ is commutative.

$(x * y) * z = \sqrt{(\sqrt{x^2 + y^2})^2 + z^2} = \sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (\sqrt{y^2 + z^2})^2} = x * (y * z),$ so $*$ is associative.

$x * e = \sqrt{x^2 + e^2} = x,$ so $e = 0.$ Further, $e * x = \sqrt{0 + x^2} = x.$ So, $*$ has an identity.

$x * x' = \sqrt{x^2 + x'^2} = e,$ so $x' = \sqrt{e^2 - x^2} \text { and } x' = -\sqrt{e^2 - x^2}$. Further, $x' * x = \sqrt{(\sqrt{e^2 - x^2})^2 + x^2} = e.$ So, every element has an inverse.

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Let $x * y = x - y.$

$x * y = x - y \neq y - x,$ so $*$ is not commutative.

$(x * y) * z = ((x - y) - z) = (x - (y - z),$ so $*$ is not associative.

$x * e = x - e = x,$ so $e = 0$. Further, $e * x = 0 - x = -x.$ So, $*$ has no identity.

$x * x' = x - x' = e,$ so $x' = x - e$. Further, $x' * x = x - e - x = -e.$ So, not every element has an inverse.

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Let $x * y = xy + 1.$

$(x * y) * z = xy+ 1 = 1 + xy,$ so $*$ is commutative.

$(x * y) * z = (xyz + z) + 1 = xyz + (z + 1),$ so $*$ is associative.

$x * e = xe + 1 = x, $ so $e = 0$. Further, $e * x = 0x + 1 = 1.$ So, $*$ has no identity.

$x * x' = xx' + 1 = e,$ so $x' = \frac {e - 1}{x}.$ Further, $x' * x = \frac {e - 1}{x} x + 1 = e$, so every element has an inverse.

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Let $x * y = max\{x, y\}.$

$(x * y) * z = max\{max\{x, y\}, z\} = max\{max \text { } x, \{y , z\}\},$ so $*$ is associative.

$x * e = max\{x, e\} = x,$ so $e = min\{x, e\}.$ Futher, $e * x = max\{min\{x, e\}, x\} = x,$ so $*$ has an identity.

$x * x' = max\{x, x'\} \neq e,$ so $x'$ is undefined. Not every element has an inverse.

Answer 1

List of mistakes and potential improvements:

Problem 1) The associativity proof is not correct, and you can use a counterexample to prove it - say $||1+1|-2|=0$, but $|1+|1-2||=2$.

Problem 2) In the associativity proof, to be technically correct, it should read $||xy|z|=|x|yz||$ instead of just having parentheses for the inside pairings. The no identity part of this could also use cleaning up: you should state more clearly why no identity exists - say be considering $x=-1$ and noting that whatever the other element $x*y\ge0>x$

Problem 3) You should check here that an inverse actually exists - you have an expression here, but you know $e=0$. When you try and find the inverse of $1$, you may have some problems with taking the square root of a negative number.

Problem 4) There should be a $\ne$ sign in your associativity proof, but your conclusion is correct.

Problem 5) There is a $z$ in commutivity case when there shouldn't be - I think this was just a typo. The associativity case is wrong: $(x*y)*z=xyz+z+1$, but $x*(y*z)=xyz+x+1$. I don't see how you concluded $e=0$ - solving for $e$ gets you $e=\frac{x-1}{x}$ which has problems in that it is dependent on $x$ - so no identity exists. Since no identity exists, it's impossible for inverses to exist.

Problem 6) The identity needs to work for an arbitrary element hence you need $e=-\infty$. A short statement of commutivity should also be included in this part.

Answer 2

For number 5: it is not associate. you did not check: $(x∗y)∗z=x∗(y∗z)$

Also, you said "$xe+1=x$, so $e=0$" well if $e=0$ then we get $1=x$ but we need $x=x$ either way, you are right there is no identity but you did not provide the right reason.

This is important: If there is no identity then there is no inverse. So you contradicted yourself in the inverse part.

Hopefully this helps!