Problem 68 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: If \(A\) and \(B\) be two sets having \(p\) and \(q\) elements respectively, where \(q>p\). Then the total number of functions from set \(A\) to set \(B\) is \(q^{p}\). Statement 2: The total number of selections of \(p\) different objects out of \(q\) objects is \({ }^{q} \mathrm{C}_{p}\) (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 . (c) Statement 1 is false, Statement 2 is true (d) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1 .

Step-by-Step Solution

Answer

(b) Statement 1 true, Statement 2 true, not a correct explanation of 1.

Step 1: Understanding Statement 1

Statement 1 claims that if there are two sets, \(A\) with \(p\) elements and \(B\) with \(q\) elements where \(q > p\), the total number of functions from \(A\) to \(B\) is \(q^p\). This is true because functions from \(A\) to \(B\) can map each of the \(p\) elements in \(A\) to any of the \(q\) elements in \(B\). Thus, there are \(q\) choices for each of \(p\) elements, making a total of \(q^p\) functions. Hence, Statement 1 is true.

Step 2: Understanding Statement 2

Statement 2 states that the total number of selections of \(p\) different objects out of \(q\) objects is \({ }^{q} \mathrm{C}_{p}\). This is a true statement as it describes the combination formula used to find the number of ways to select \(p\) objects from a set of \(q\) without regard to the order of selection. Hence, Statement 2 is true.

Step 3: Evaluating the Relationship

To determine if Statement 2 is a correct explanation of Statement 1, we analyze their relationship. Statement 1 involves mapping elements from one set to another (function mapping), while Statement 2 involves the selection of elements without order. The true explanation for the number of functions in Statement 1 is based on counting principles, while Statement 2 is about combinations and selections. Thus, although both statements are true, Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arrangement, and combinations within a set. It helps answer questions like "How many ways can objects be chosen or ordered?" This is crucial in developing problem-solving skills. Consider the concept of combinations, which involve selecting items from a larger set without regard to order. When we say, "we have 5 fruits and want to choose 3," we're using combinations. Mathematically, this is represented by \({ }^{5} \mathrm{C}_{3}\). Here, the combination formula is used, which is \[^{n}C_k = \frac{n!}{k!(n-k)!}\]where \( n! \) (factorial) represents the product of an integer and all the integers below it. This formula provides the number of ways to choose \( k \) objects from \( n \) objects. In essence, combinatorics is about finding the possible configurations or arrangements within a certain set of parameters. By understanding this, students gain a deeper insight into problem-solving across various fields like computer science, physics, and statistics.

Set Theory

Set theory is the study of collections of objects, known as sets. A set is a fundamental concept in mathematics and is used to group objects. For instance, the set of natural numbers \( \{1, 2, 3, \ldots \} \) is one example. Sets are often described using curly braces and can sometimes include infinite elements.
Set theory is crucial because it forms the foundation for various mathematical disciplines. It helps in understanding relationships among different collections of objects. There are specific operations on sets, such as union, intersection, and difference. These help in manipulating and analyzing sets:

• Union: Combines all elements from two sets.
• Intersection: Takes only the common elements from two sets.
• Difference: Consists of elements present in one set but not in the other.

Through set theory, we learn to conceptualize and structure mathematical ideas, which is essential in fields like computer science, where data and relationships among data are extensively studied.

Function Mapping

Function mapping refers to the relationship between two sets where each element in the first set is paired with exactly one element in the second set. This is called a function. Think of a function as a kind of mathematical 'machine' where you input a value (from the domain set) and get an output (from the codomain set).
In the context of sets, if we have set \( A \) with \( p \) elements and set \( B \) with \( q \) elements where \( q > p \), then we can create \( q^p \) different functions. Each element in \( A \) can connect or "map" to any element in \( B\), providing \( q \) choices per each of the \( p \) elements.

• Domain: The set of all possible inputs.
• Codomain: The set of possible outputs.
• Range: The actual set of outputs produced by the function.

Understanding function mapping is essential for many fields such as physics, engineering, and computer science. It allows for conceptualizing relationships and transformations between different quantities and datasets.