Is the following relation considered symmetric?

Question

i'm having a little misunderstanding about how to determinate if a relation is symmetric ( in the case of a function)

So i have : $\{ \langle x,y\rangle \in\Bbb N^2 \mid x = y + 10\}$ , so the definition of the relation symmetric is saying $x R y \to y R x$ which in our case the inverse relation would be $\{ \langle x,y\rangle \in\Bbb N^2 \mid x = y + 10\}$ OR $\{ \langle x,y\rangle \in\Bbb N^2 \mid y - 10 = x \}$

I'm wondering how do i check this mathematically ? do i just choose a random pair $\langle x,y\rangle$ and place it in the inverse relation ex: for $\langle 0,10\rangle$ we would have $xRy : 0 = 10+ 10$ and for $yRx : 10=10 + 0 $ so the two are not equal so its not symmetric ?

Thank you

Answer 1

To demonstrate that a relation is not symmetric, it does suffice to find one counter example.

  So indeed demonstrating that we have $10\mathcal R0$ and $\lnot 0\mathcal R10$ does the job.

More generally, it is clear that where ever $x=y+10$, there $y=x-10$, and so $y\neq x+10$.

Answer 2

To figure out if it's symmetric, you check if $ \langle x, y \rangle $ and $ \langle y, x \rangle $ are in it for all $ x $ and $ y $.

So, let's do it with arbitrary $ x $s and $ y $s:

$$ \begin{align} x \mathrm{R} y &\wedge y \mathrm{R} x \\ x = y + 10 &\wedge y = x + 10 \\ x = y + 10 &\wedge x = y - 10 \\ x &= x \\ y + 10 &= y - 10 \\ 10 &= -10 \\ &\textrm{False} \end{align} $$

By universal introduction, we see that there are no values at all of $ x $ and $ y $ for which the relation is symmetric.