Let A={1,2,3} and B={4,5,6}. Which one of the following functions is bijective?

The correct option is f = {(1,4), (2,5), (3,6)}.

To understand why this function is bijective, we need to analyze the definitions of a function, one-to-one (injective) functions, and onto (surjective) functions.
Let $A=\{1,2,3\}$ be the domain and $B=\{4,5,6\}$ be the codomain.

1. Definition of a Function:
A relation $f$ from $A$ to $B$ is a function if every element in the domain $A$ is mapped to exactly one element in the codomain $B$.

Let's check the given options:

• For $f=\left\{\right(2,4),(2,5),(2,6\left)\right\}$, the element 2 is mapped to three different values (4, 5, and 6), and elements 1 and 3 are not mapped to any values. Thus, this is not a function.
• For $f=\left\{\right(1,5),(2,4),(3,4\left)\right\}$, each element of $A$ is mapped to exactly one element of $B$. This is a valid function.
• For $f=\left\{\right(1,4),(1,5),(1,6\left)\right\}$, the element 1 is mapped to multiple values, and elements 2 and 3 are unmapped. Thus, this is not a function.
• For $f=\left\{\right(1,4),(2,5),(3,6\left)\right\}$, each element of $A$ (1, 2, and 3) is mapped to exactly one element of $B$. This is a valid function.

2. Bijective Function Conditions:
A function is bijective if it is both injective (one-to-one) and surjective (onto).

Injectivity (One-to-One): A function $f:A\to B$ is injective if distinct elements in $A$ map to distinct elements in $B$.
For $f=\left\{\right(1,4),(2,5),(3,6\left)\right\}$:
$f\left(1\right)=4$, $f\left(2\right)=5$, and $f\left(3\right)=6$.
Since no two elements in the domain map to the same image in the codomain, the function is injective.

Surjectivity (Onto): A function $f:A\to B$ is surjective if the range of the function is equal to the codomain $B$.
The codomain is $B=\{4,5,6\}$.
The range of $f$ is the set of all second components in the ordered pairs: $\{4,5,6\}$.
Since the Range = Codomain, the function is surjective.

Because the function $f=\left\{\right(1,4),(2,5),(3,6\left)\right\}$ is both injective and surjective, it is bijective.