Let A be a set consisting of 10 elements. number of non-empty relations from A to A that are reflexi

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1.Relation and Function

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Let $A$ be a set consisting of $10$ elements. The number of non-empty relations from $A$ to $A$ that are reflexive but not symmetric is

A

$2^{89}-1$

B

$2^{89}-2^{45}$

C

$2^{45}-1$

D

$2^{90}-2^{45}$

(KVPY-2020)

Solution

(d)

Since, $A \times A$ contains $100$ ordered pairs $(a, b)$ out of which $10$ ordered pairs are such that $a=b$.

For a reflexive relation $(a, a)$ must be present and others have a choice of to be present or not.

So, number of reflexive relations $=2^{90}$.

For a symmetric relation if $(a, b)$ is present then $(b, a)$ is also present

(where $a \neq b$ ). There are $45$ such pairs of ordered pairs.

So, number of reflexive relations which are also symmetric $=2^{45}$

$\therefore$ Required number of relations $=2^{90}-2^{45}$.

Standard 12

Mathematics

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