Let \(A = \{1, 2, 3\}\). Match the functions in List-I with their types in List-II.
| List-I | List-II | ||
| (A) | \(f = \{(1, a), (2, b), (3, c)\}\) from A to \(\{a, b, c\}\) | (I) | One-one and onto |
| (B) | \(f = \{(1, a), (2, a), (3, b)\}\) from A to \(\{a, b\}\) | (II) | One-one but not onto |
| (C) | \(f = \{(1, b), (2, c), (3, d)\}\) from A to \(\{a, b, c, d\}\) | (III) | Onto but not one-one |
| (D) | \(f = \{(1, a), (2, a), (3, b)\}\) from A to \(\{a, b, c\}\) | (IV) | Neither one-one nor onto |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(I), (B)-(III), (C)-(II), (D)-(IV)
A function is one-one when no two elements share an image, and onto when every element of the codomain is used as an image.
(A) \(f = \{(1, a), (2, b), (3, c)\}\) from A to \(\{a, b, c\}\)
This matches (I) One-one and onto.
the images a, b, c are all different
they use up the whole codomain \(\{a, b, c\}\)
(B) \(f = \{(1, a), (2, a), (3, b)\}\) from A to \(\{a, b\}\)
This matches (III) Onto but not one-one.
\(f(1) = f(2) = a\)
both a and b are images
(C) \(f = \{(1, b), (2, c), (3, d)\}\) from A to \(\{a, b, c, d\}\)
This matches (II) One-one but not onto.
the images b, c, d are all different
a is not an image
(D) \(f = \{(1, a), (2, a), (3, b)\}\) from A to \(\{a, b, c\}\)
This matches (IV) Neither one-one nor onto.
\(f(1) = f(2) = a\)
c is not an image