Proving or disproving statements about operations with integers

Question

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

Answer 1

Hint

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 : $a^2 + b^2 = b^2 + a^2$.

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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...

Answer 2

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.