Let f : A ----> B be a function.
The function f is called an onto function, if every element in B has a pre-image in A.
That is, in B all the elements will be involved in mapping.
An onto function is also called a surjective function.
The figure given below represents a onto function.
Problem 1 :
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)}
Verify whether f is a function. if so, what type of function is f ?
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.
Therefore, f is onto or surjective function.
Problem 2 :
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 surjective ? 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.
The element "7" in B has no pre-image in A.
Because the element "7" has no pre-image, f is not onto or surjective function.
Problem 3 :
Let f : A ----> B. A, B and f are defined as
A = {1, 2, 3, 4}
B = {5, 6, 7, 8}
f = {(1, 8), (2, 6), (3, 5), (4, 7)}
Is f surjective ? 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.
Every element of B has a pre- image in A.
Therefore, f is onto or surjective function.
One to one or Injective function
One to one and Onto or Bijective function
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