The number of relations, defined on the set ${a, b, c, d}$, which are both reflexive and symmetric, is equal to:

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:
$R = \{(x, y) : x \text{ is exactly } 7 \, cm \text{ taller than } y\}$

Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$ is an equivalence relation.

Let $A = \{1, 2, 3\}$. The number of equivalence relations on $A$ containing $(1, 2)$ is . . . . . . .

Let $T$ be the set of all triangles in a Euclidean plane and a relation $R$ on $T$ is defined as $aRb$ if and only if $a \sim b$ (where $a \sim b$ denotes that triangle $a$ is similar to triangle $b$) for all $a, b \in T$. Then $R$ is:

Let $A = \{1, 2, 3, 4\}$ and $R$ be a relation on $A$,given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\}$. Then $R$ is: