Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$ be denoted by $S$,such that $R$ is reflexive and symmetric,and $R$ contains exactly $10$ elements. Then the number of elements in $S$ is $...$ .

On the set of integers $Z$,a relation $S$ is defined as: $S = \{(x, y) \in Z \times Z : |x - y| < 1\}$. Which of the following is true about $S$?

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a, b) | a \ge b, a, b \in R \}$. Then ${R_1}$ 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

Show that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R =\{(a, b):|a-b| \text{ is even}\}$ is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other,but no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

Let $A$ be the non-void set of the children in a family. The relation '$x$ is a brother of $y$' on $A$ is