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CBSE Sample Paper 2023-24 · Section A · 1 mark

Q20: Is {(1,x),(2,y),(3,z)}\{(1, x), (2, y), (3, z)\} a bijection from {1,2,3,4}\{1, 2, 3, 4\}⁠?

The question

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 f:{1,2,3,4}→{x,y,z,p}f : \{1, 2, 3, 4\} \to \{x, y, z, p\} defined by f={(1,x),(2,y),(3,z)}f = \{(1, x), (2, y), (3, z)\} is a bijective function.

REASON (R): The function f:{1,2,3}→{x,y,z,p}f : \{1, 2, 3\} \to \{x, y, z, p\} such that f={(1,x),(2,y),(3,z)}f = \{(1, x), (2, y), (3, z)\} is one-one.

Exam answer

In (A), the element 4 of the domain has no image under ff⁠, so ff 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 {1,2,3,4}\{1, 2, 3, 4\}⁠, but ff 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 xx⁠, yy⁠, zz⁠, so ff is one-one. So Reason (R) is true.

Why?

One-one means that different elements have different images: f(a)=f(b)f(a) = f(b) only when a=ba = b⁠.

Here f(1)=xf(1) = x⁠, f(2)=yf(2) = y⁠, f(3)=zf(3) = z⁠, and no two of these are equal.

Watch out: this ff is not onto, since pp 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.

Where this comes from

Chapter 1, Relations and Functions: One-one, onto and bijective functions.

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