So I am facing some issues determining the right properties for: $ xRy\;if\,\sin^2(x) + \cos^2(y) = 1 $. (On real numbers)
Obviously this one is reflexive as $\sin^2(x) + \cos^2(x) = 1 $ is a basic trigonometric identity. Where I'm blocking though is when I try to determine if it's symmetric, antisymmetric and transitive.
I kind of have the feeling that it is symmetric since when $(a,b)$ is part of the relation, so is $(b,a)$ (as long as $a=b$). This would also mean it is antisymmetric. And finally it should be transitive since the only $(x,y)$ couples that this relation takes are couples where $x=y$ and therefore $y = z$.
The answer apparently is what I just said except for the antisymmetric part (according to my TA). I was wondering what it is that I am not getting about the asymmetry part.
Thanks in advance!
Your reasoning for transitive is wrong because $\sin(0)^2+\cos(2\pi)^2=1$ and $0 \ne 2\pi$. Just expand the definition of transitivity of $R$ on $\mathbb{C}$ as follows.
For any $x,y,z \in \mathbb{C}$:
If $x R y$ and $y R z$:
... [Expand the definition of $x R y$ and $y R z$ here.]
... [Manipulate what you have here until you get the definition of $x R z$.]
$x R z$
Therefore $R$ is transitive on $\mathbb{C}$
Same for symmetric:
For any $x,y \in \mathbb{C}$:
If $x R y$:
... [Expand the definition of $x R y$ here.]
... [Manipulate what you have here until you get the definition of $y R x$.]
$y R x$
Therefore $R$ is symmetric on $\mathbb{C}$