$A$ relation $R$ on a non-empty set $A$ is an equivalence relation if $R$ is:

If $R$ and $R^1$ are equivalence relations on a set $A$, then which of the following is also an equivalence relation?

Let $L$ be the set of all lines in the $XY$ plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \text{ is parallel to } L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.

For any two real numbers $\theta$ and $\phi$, we define $\theta R \phi$ if and only if $\sec^{2} \theta - \tan^{2} \phi = 1$. The relation $R$ is

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are said to be related by the relation $R$ if and only if $l_1$ is parallel to $l_2$. Then the relation $R$ is

Let $L$ be the set of all lines in a plane and $R$ be the relation in $L$ defined as $R = \{(L_{1}, L_{2}) : L_{1} \text{ is perpendicular to } L_{2}\}$. Show that $R$ is symmetric but neither reflexive nor transitive.