INTO FUNCTION

Let f : A ----> B be a function.

There exists even a single element in B having no pre-image in A, then f is said to be an into function.

The figure given below represents a one-one function.

Solved Problems

Problem 1 :

Let f : A ----> B. A, B and f are defined as

A = {1, 2, 3}

B = {5, 6, 7, 8}

f = {(1, 5), (2, 8), (3, 6)}

Is f into function? Explain.

Solution :

Write the elements of f (ordered pairs) using arrow diagram as shown below

In the above arrow diagram, all the elements of A have images in B and every element of A has a unique image.

That is, no element of A has more than one image.

So, f is a function.

There exists an element 7 in B having no pre-image in A.

Therefore, f is into function.

Problem 2 :

Let f : X ----> Y. X, Y and f are defined as

X = {a, b, c, d}

Y = {d, e, f}

f = {(a, e), (b, f), (c, e), (d, d)}

Is f into function ? Explain.

In the above arrow diagram, all the elements of X have images in Y and every element of X has a unique image.

That is, no element of X has more than one image.

So, f is a function.

Every element of Y has a pre-image in X.

So, f is not into function.