In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation,it is sufficient,if $R$

Let a relation $R$ on $N \times N$ be defined as: $(x_1, y_1) R (x_2, y_2)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:
$(I)$ $R$ is reflexive but not symmetric.
$(II)$ $R$ is transitive.
Then which one of the following is true?

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R = \{(x, y) \in N \times N : x^{3}-3x^{2}y-xy^{2}+3y^{3}=0\}$. Then the relation $R$ is:

Let $R$ be a relation from $Q$ to $Q$ defined by $R = \{(a, b) : a, b \in Q \text{ and } a - b \in Z \}$. Show that $(a, a) \in R$ for all $a \in Q$.

Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_{1}, P_{2}) : P_{1} \text{ and } P_{2} \text{ have the same number of sides}\}$,is an equivalence relation. What is the set of all elements in $A$ related to the right-angled triangle $T$ with sides $3, 4, \text{ and } 5$?

Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by $R = \{(x, y): x \text{ is the father of } y\}$.