CBSE Sample Paper 2020-21 · Part A: Section I · 1 mark

Q1: Is $f(x) = x^3$ one-one? Or: the number of reflexive relations on a 3-element set

Sample 2020-21Ch 1 · Relations and Functions

The question

(a) Check whether the function $f : R \to R$ defined as $f(x) = x^3$ is one-one or not.

(b) How many reflexive relations are possible in a set A whose $n(A) = 3$ ⁠.

Exam answer

(a)

Let $f(x_1) = f(x_2)$ for some $x_1, x_2 \in R$ ⁠.

Why?

A function is one-one when equal outputs can only come from equal inputs:

f(x_1) = f(x_2) \Rightarrow x_1 = x_2

So we start from two equal outputs and try to reach $x_1 = x_2$ ⁠.

\begin{aligned} &\Rightarrow (x_1)^3 = (x_2)^3 \\ &\Rightarrow x_1 = x_2 \end{aligned}

(a real number has only one real cube root)

Why?

Taking the cube root of both sides is safe, because cubing keeps the order: if $x_1 < x_2$ ⁠, then $x_1^3 < x_2^3$ ⁠. So two different real numbers never have the same cube. Even the sign survives: $2^3 = 8$ but $(-2)^3 = -8$ ⁠.

Watch out: this fails for squares. $2^2 = (-2)^2 = 4$ ⁠, which is why $x^2$ is not one-one on R.

Hence $f(x)$ is one-one.

(b)

A reflexive relation on A must contain the 3 pairs $(a, a)$ ⁠, $a \in A$ ⁠. Each of the other $3^2 - 3 = 6$ pairs of $A \times A$ may or may not be in it.

Why?

A relation on A is any subset of $A \times A$ ⁠, which has $3 \times 3 = 9$ ordered pairs.

Reflexive means $(a, a) \in R$ for every $a \in A$ ⁠. So the 3 pairs $(a, a)$ are forced in, and only the other 6 pairs are free to choose.

So the number of reflexive relations is

2 \times 2 \times \cdots \times 2\ (6 \text{ times}) = 2^6

Why?

Each free pair has 2 choices: in the relation or out of it. The choices are independent, so they multiply (the multiplication principle).

2^6 = 64

$\therefore$ $2^6 = 64$ reflexive relations are possible.

Where this comes from

Chapter 1, Relations and Functions: One-one functions; counting reflexive relations.

Question type: (a) Prove one-one and onto; (b) Counting questions

How to spot it: click to see

(a) Type 5. Prove one-one and onto: the bijective function

Spot it: "show that $f$ is one-one and onto", or "is $f$ bijective?", with a formula like $f(x) = 3 - 4x$ ⁠.

Here: Only one-one is asked: $x_1^3 = x_2^3$ gives $x_1 = x_2$ ⁠, since a real number has only one real cube root.

(b) Type 8. Counting questions: list the possibilities

Spot it: "the number of relations (or functions) …" on a small set like $\{1, 2, 3\}$ ⁠, usually an MCQ.

Here: The 3 pairs $(a, a)$ are forced in. Each of the other $9 - 3 = 6$ pairs of $A \times A$ is in or out, which gives $2^6 = 64$ ⁠.

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Q1: Is f(x)=x3f(x) = x^3f(x)=x3 one-one? Or: the number of reflexive relations on a 3-element set

Q1: Is $f(x) = x^3$ one-one? Or: the number of reflexive relations on a 3-element set