Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(S\) be the set of all functions from the set \(A\) to the set A. If \(n(A)=k\) then \(n(S)\) is (A) \(k !\) (B) \(k^{k}\) (C) \(2^{k}-1\) (d) \(2^{k}\)

Step-by-Step Solution

Answer

The number of functions is \( k^k \), which corresponds to option (B).

Step 1: Understanding the Set Definitions

The problem asks about the number of functions from a set \( A \) to itself, i.e., \( S = A^A \). Here, \( A \) is a finite set with \( n(A)=k \) elements.

Step 2: Identifying a Single Function

A function maps every element in set \( A \) to an element in set \( A \). Since \( n(A)=k \), each element of \( A \) can be mapped to any of the \( k \) elements in \( A \).

Step 3: Calculating the Total Number of Functions

For each of the \( k \) elements in \( A \), we have \( k \) choices. Therefore, the total number of different functions is calculated as the product of possibilities, which is \( k^k \).

Step 4: Finding the Correct Option

The number of functions from \( A \) to itself, given \( n(A) = k \), is \( k^k \). Referring back to the options, option (B) corresponds to \( k^k \).

Key Concepts

Set Theory

Set theory is a fundamental area of mathematics focused on the study of collections of objects, known as sets. A set is a well-defined collection of distinct objects, which can be anything from numbers to letters or even other sets. The sets are often denoted by capital letters like \( A, B, C \), and they can be finite or infinite based on the number of elements they contain.

• A finite set has a countable number of elements.
• An infinite set cannot be counted completely, like the set of all natural numbers.

Set theory provides the basics for more complex mathematical concepts and deals with operations like union, intersection, and difference of sets. Understanding how to work with sets and using operations on them is crucial in many areas of mathematics.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of objects. It is essential in determining how different configurations or arrangements can be made from a particular set.

• It helps in calculating probabilities when combined with probability theory.
• Uses include determining the number of ways to arrange a set of objects, or how many subsets can be formed from a set.

In the context of functions from one set to another, combinatorics comes into play by counting the total number of possible mappings. Each function represents a unique arrangement of elements from the set being mapped to. Thus, knowledge of combinatorial principles is key when determining the number of functions possible, as seen in the solution where we found \( k^k \) possible functions for a set of size \( k \).

Function Mapping

Function mapping is a process in mathematics where each element of one set, known as the domain, is paired with an element of another set called the codomain. This pairing defines a function, and if the domain and codomain are the same set, it is known as a mapping from a set to itself.

• A function \( f \) from a set \( A \) to a set \( B \) is denoted as \( f: A \rightarrow B \).
• If every element of \( A \) is mapped to exactly one element of \( B \), then \( f \) is considered a well-defined function.

In our exercise, each element of set \( A \) can map to any other element, including itself. This means that when set \( A \) has \( k \) elements, we have \( k \) choices per mapping, resulting in \( k^k \) different functions from \( A \) to itself.

Finite Sets

Finite sets are sets that contain a limited number of elements. The cardinality or size of a finite set is the number of elements it includes, denoted as \( n(A) \) for a set \( A \). Finite sets are crucial in making calculations and solving problems in mathematics, as they enable concrete counting and enumeration.

• In the context of our exercise, \( A \) has \( k \) elements, making it a finite set.
• The concept of finite sets is used to determine the total number of functions possible, by counting how each of the finite elements can appear in pairs or functions.

Finite sets facilitate the use of formulas and combinatorial methods to derive solutions, like calculating the number of functions or other configurations, as seen in the exercise where \( S \) is derived as \( A^A \), leading to \( k^k \) possible functions.