Page 19 - Mathematics Class - XII

3. Join the board pins with the help of threads or yarns or rubber band as shown in Fig. (c).
A B

1 •
• x
15 cm 2 • • y 12 cm
• z
3 •
4 • • u
• v
5 cm 5 cm
Fig. (c)

DEMONSTRATION
1. Take the set A = {1, 2, 3, 4}
2. Take the set B = {x, y, z, u, v}.

3. Join elements of A to the elements of B as shown in Fig. (c).

OBSERVATION
1. The image of element 1 of set A is y in B.
2. The image of element 2 of set A is u in B.
3. The image of element 3 of set A is v in B.

4. The image of element 4 of set A is z in B.
5. The pre-image of element y of B is 1 in A.
6. The pre-image of element z of B is 4 in A.
7. The pre-image of element u of B is 2 in A.

8. The pre-image of element v of B is 3 in A.
9. Elements x of B has no pre-image in A. So, the function f is not onto.
10. Since the images of distinct element of set A are distinct, so the function f is one-one.

11. From observation 9 and 10, we observe that the function which is one-one but not onto.

CONCLUSION
From the above activity we have demonstrated a function which is one-one but not onto.

APPLICATION
This activity can be used to demonstrate the concept of one-one and onto functions.

Note: Demonstrate the same activity by changing the number of the elements of the sets A and B.

Knowledge Booster
Domain of a function can be a specific set of numbers. The domain can also be all numbers except one or two for which
1
the function doesn't work. For example, the domain for the function f (x) = is all numbers except 4, because
4 − x
when you input 4, the denominator is 0, and the result is undefined. The domain for 1 , on the other hand, is all
9 − x 2
numbers except +3 and –3 because the square of both of these numbers is 9, making the result undefined.

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