**One-one Functions**

Let A and B be sets and *f* is a function from A to B. The function *f* is called a * one-one function *or an

*if the following holds:*

**injective function**f(a) = f(b) ⇒ a = b ………………………………………………………………………………………………………………………………… (1)

Loosely speaking, the function is one-one if no two different elements of A have the same image in B.

**Example 1: (not an injective function)**

Let f: be defined by f(x) = x^{2}.

Note that f(-5) = (-5)^{2} = 25 and f(5) = 5^{2} = 25.

In this case, f(-5) = f(5), but -5 ≠ 5 .

In this example, the condition (1) is not satisfied, thus we conclude that f is not an injective function.

**Example 2: (injective function)**

Let g: be defined by g(x) = x + 10.

We claim that g is an injective function.

To prove this, let g(a) = g(b).

As a consequence, a + 10 = b + 10.

Subtract 10 from both sides of the equation. Then we have a = b.

We have proved that g(a) = g(b) ⇒ a = b.

By (1), we conclude that g is an injective function.

**Example 3: (not a one-one function)**

Let h: be defined by h(x) = |x|.

Note that h(3) = |3| = 3 and h(-3) = |-3| = 3.

In this case, h(3) = h(-3) but -3 ≠ 3.

The condition (1) is violated, so we conclude that h is not a one-one function.

**Example 4 (one-one function)**

Let *k*: be defined by

We claim that *k* is a one-one function.

To prove this, let *k*(a) = *k*(b).

Consequently,

By multiplying both sides of the equation by ab, we have a = b.

We have proved that *k*(a) = *k*(b) ⇒ a = b.

By (1), it follows that *k* is a one-one function.

**Surjective Functions**

Let A and B be sets and f is a function from A to B. The function f is called a * surjective function *or an

*if the following holds:*

**onto function**∀b ∈ B ∃a ∈ A ∋ f(a) = b ………………………………………………………………………………………………………………………………………………………… (2)

Stated another way, a function f is said to be surjective if for every b ∈ B there is some a ∈ A such that f(a) = b. Loosely speaking, condition (2) means: “to each element in B there is assigned an element of A”.

**Example 5 (not a surjective function)**

Let f: be defined by f(x) = x^{2}.

f is not a surjective function because -3 ∈ but there is no such that f(*a*) = *a*^{2} = -3.

In this case, condition (2) is not satisfied. So we conclude that f is not a surjective function.

**Example 6 (surjective function)**

Let g: be defined by g(x) = x + 10

We claim that g is a surjective function

To prove it, let .

Choose *a* = (*b* – 10) ∈ .

Consequently, g(*a*) = (*b* – 10) + 10 = *b*.

We have proved that for every there is some such that g(*a*) = *b*. By (2), we can conclude that g is a surjective function.

**Example 7 (not a surjective function)**

Let *k *: be defined by .

We claim that *k* is not a surjective function.

To prove it, choose 0 ∈ . For any it is impossible that .

We have proved that there is a such that for every *k*(*a*) ≠ *b*. Hence, *k* is not surjective.

**Example 8 (surjective function)**

Let *v *: be defined by .

We claim that v is a surjective function.

To prove it, let . It is obvious that *b* ≠ 0. Therefore, we can select . This choice implies .

We have proved that for every there is some such that v(*a*) = *b*. By (2), we can conclude that *v* is a surjective function.