Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is :

Let $\mathrm{S}=\mathbf{N} \cup\{0\}$. Define a relation R from S to $\mathbf{R}$ by :

$$ \mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\} . $$

Then, the sum of all the elements in the range of $R$ is equal to :

Define a relation R on the interval $ \left[0, \frac{\pi}{2}\right) $ by $ x $ R $ y $ if and only if $ \sec^2x - \tan^2y = 1 $. Then R is :

The relation $R=\{(x, y): x, y \in \mathbb{Z}$ and $x+y$ is even $\}$ is: