The total number of relations that exist from a set $A$ with $m$ elements into the set $A \times A$ is:

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — None of these

Detailed Solution

A relation from $A$ to $B$ is any subset of $A \times B$ . Here $B = A \times A$ has $m^{2}$ elements.

$A \times (A \times A)$ has $m \cdot m^{2} = m^{3}$ elements, so the number of relations is $2^{m^{3}}$ .

$2^{m^{3}}$ is not $m^{2}$ , $m^{3}$ or $m$ , so the answer is None of these.

✓ Check

For

m = 1

the product has one element, giving 2 relations (the empty set and the whole set), and

2^{1^{3}} = 2

⚠ Common trap

(B)

m^{3}

counts the ordered pairs in

A \times (A \times A)

, not the subsets (relations) made from them.

Quick tip

$relations = 2^{m \cdot m^{2}} = 2^{m^{3}}$

💬 Discussions on this question

Ask a doubt ›

Still stuck on this question? Ask our faculty and fellow students — answers appear here.

Ask a doubt

More Relations & Mappings Questions

View all ›

Relations & Mappings — Preparation

This question tests your understanding of Relations & Mappings, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A relation from set A to set B is a subset of the Cartesian product A x B. A mapping, or function, is a special relation in which every element of A is related to exactly one element of B. This topic covers types of relations (reflexive, symmetric, transitive and equivalence relations), domain, codomain and range, and types of functions such as one-one (injective), onto (surjective) and bijective functions, along with composition and inverse functions. Exam questions test identifying the type of a given relation or function, counting the number of relations or functions between two sets, and finding the domain or range of a function. Useful counting facts: if A has m elements and B has n elements, there are 2^(mn) relations and n^m functions from A to B. Practise the relations and mappings questions below, each with a step-by-step solution.

‹ All Relations & Mappings questions Browse all topics ›

The total number of relations that exist from a set A with m elements into the set A×A is:

💬 Discussions on this question

More Relations & Mappings Questions

More Quantitative Aptitude Questions

Relations & Mappings — Preparation

The total number of relations that exist from a set $A$ with $m$ elements into the set $A \times A$ is: