Q20: Is a bijection from ?
In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices.
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are true but (R) is not the correct explanation of (A).
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true.
ASSERTION (A): The relation defined by is a bijective function.
REASON (R): The function such that is one-one.
In (A), the element 4 of the domain has no image under , so is not a function. So Assertion (A) is false.
Why?
A relation is a function only when every element of its domain has exactly one image.
Here the domain is , but has no pair starting with 4. A relation that is not a function cannot be a bijective function.
In (R), the elements 1, 2, 3 have the different images , , , so is one-one. So Reason (R) is true.
Why?
One-one means that different elements have different images: only when .
Here , , , and no two of these are equal.
Watch out: this is not onto, since has no pre-image. But (R) claims only one-one, so it is still true.
(A) is false and (R) is true, which is code (d).
Answer: (d) (A) is false but (R) is true.
Chapter 1, Relations and Functions: One-one, onto and bijective functions.
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