I'm really stuck with this one and I'm thankful for any help.

Consider the following operations on the set of integers:

$\hspace{8em} a\star b := a^2 + b^2 \hspace{5em} a\diamond b := a+b+2ab$

Prove or disprove the following statements:

a.) The relation is associative

b.) The relation is commutative
I do know that addition or multiplication relations are associative but I can't conclude that $a\star b$ or $a\diamond b$ are?

c.) There exists an identity element / neutral element
d.) For every integer there exists an inverse element


You have to understand the various definitions. For example, to check associativity, ask yourself what needs to be verified. In case of the operation $\star$ we need to check if

$$(a \star b) \star c=a \star (b \star c)?$$

Start with left side: \begin{align*} (a \star b) \star c & = (a \star b)^2 + c^2\\ & = (a^2+b^2)^2+c^2. \end{align*} whereas the right side will be \begin{align*} a \star (b \star c) & = a \star (b^2 + c^2)\\ & = a^2+(b^2+c^2)^2. \end{align*} It can be easily seen that these two expressions need not be equal for all integers. Hence the operation is NOT associative.

Now try other statements.



You have to recall that a (binary) mathematical "operation" is a relation, i.e. a set of pairs; thus, applying the set definition of relation to the newly defined "operation" $⋆$, we have that :

$(a,b) \in ⋆$ iff $a^2 + b^2$.

We say that a mathematical operation $\circ$ is commutative when :

$x \circ y = y \circ x\quad\text{for all }x,y$.

Thus, we have to show that :

$a⋆b = b⋆a$

and this is easily proved due to the fact that : $a^2 + b^2 = b^2 + a^2$.

For associativity, you have to check if :

$(a⋆b)⋆c = a⋆(b⋆c)$

or not.

The LHS is : $(a^2+b^2)^2+c^2$, while the RHS is : $a^2+(b^2+c^2)^2$ and we simply have to check if they are equal or not...


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