The number of relations $R$ from an $m$-element set $A$ to an $n$-element set $B$ satisfying the condition $(a, b_1) \in R, (a, b_2) \in R \Rightarrow b_1 = b_2$ for $a \in A, b_1, b_2 \in B$ is

Which of the following is $NOT$ a function?

The relation $f$ is defined by $f(x) = \begin{cases} x^2, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases}$. The relation $g$ is defined by $g(x) = \begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases}$. Show that $f$ is a function and $g$ is not a function.

Examine the following relation and state whether it is a function or not,giving reasons:
$R = \{(2, 1), (3, 1), (4, 2)\}$

Examine the following relation and state whether it is a function or not,giving reasons:
$R = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)\}$

Let $A = \{1, 2, 3, 4\}$,$B = \{1, 5, 9, 11, 15, 16\}$ and $f = \{(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)\}$. Is $f$ a function from $A$ to $B$? Justify your answer.