For a set $A = \{1, 2, 3\}$,a relation $R = \{(1, 2), (2, 3)\}$ is defined. What is the minimum number of ordered pairs that must be added to $R$ to make it an equivalence relation?

Let $A = \{2, 3, 5, 7, 9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x \le 3y$. Let $l$ be the number of elements in $R$,and $m$ be the minimum number of elements required to be added in $R$ to make it a symmetric relation. Then $l + m$ is equal to:

Let $A = \{1, 2, 3\}$. What is the total number of relations defined on $A$?

For real numbers $x$ and $y$,we define the relation $R$ as $xRy$ if $x - y + \sqrt{2}$ is an irrational number. Then the relation $R$ is:

Let $R$ be a relation defined on $N \times N$ by $(a, b) R(c, d) \Leftrightarrow a(b + d) = c(b + d)$ is incorrect,the correct relation is $(a, b) R(c, d) \Leftrightarrow ad = bc$. Given the relation $(a, b) R(c, d) \Leftrightarrow a(b + d) = c(a + d)$ is not standard,let us analyze the relation $(a, b) R(c, d) \Leftrightarrow ad = bc$. Then $R$ is:

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