Let $X$ be a non-void set. If $\rho_1$ and $\rho_2$ are transitive relations on $X$, then which of the following is true?

Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_{1}, T_{2}) : T_{1} \text{ is similar to } T_{2}\}$,is an equivalence relation. Consider three right-angled triangles $T_{1}$ with sides $3, 4, 5$,$T_{2}$ with sides $5, 12, 13$,and $T_{3}$ with sides $6, 8, 10$. Which triangles among $T_{1}, T_{2}$,and $T_{3}$ are related?

Let $n$ be a fixed positive integer. The relation $R$ on the set of integers $Z$ is defined by $aRb \Leftrightarrow n | (a - b)$. Then $R$ is:

If $A = \{x \in Z^+ : x < 10\}$ and $x$ is a multiple of $3$ or $4$,where $Z^+$ is the set of positive integers,then the total number of symmetric relations on $A$ is

Consider the relations $R_1$ and $R_2$ defined as $a R_1 b \Leftrightarrow a^2+b^2=1$ for all $a, b \in R$ and $(a, b) R_2 (c, d) \Leftrightarrow a+d=b+c$ for all $(a, b), (c, d) \in N \times N$. Then:

Let $X$ be a family of sets and $R$ be a relation on $X$ defined by '$A$ is disjoint from $B$'. Then $R$ is