Problem 1 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Set \(A\) has 3 elements and set \(B\) has 4 elements, the number of injections that can be defined from \(A\) to \(B\) is \(\quad\) [UPSEAT-2001] (a) 144 (b) 12 (c) 24 (d) 64

Step-by-Step Solution

Answer

The number of injections is 24 (option c).

Step 1: Understanding Injections

An injection (or one-to-one function) from set \(A\) to set \(B\) means that each element from \(A\) must map to a unique element in \(B\), and no two elements from \(A\) map to the same element in \(B\). Since set \(A\) has 3 elements and set \(B\) has 4 elements, we are looking to count how many different ways we can assign unique images in \(B\) to each element in \(A\).

Step 2: Counting the Number of Choices for Mappings

To define an injective function, the first element from \(A\) can be mapped to any of the 4 elements in \(B\). Once it's chosen, the second element from \(A\) can be mapped to any of the remaining 3 elements in \(B\), because we cannot reuse the first element's image. Finally, the third element from \(A\) can be mapped to the remaining 2 elements in \(B\).

Step 3: Calculation of Injections

The total number of injective functions is the product of choices made at each step: First element choice (4 options), second element choice (3 options), and third element choice (2 options). This can be represented mathematically as \(4 \times 3 \times 2\).

Step 4: Final Calculation and Result

Calculating \(4 \times 3 \times 2 = 24\). Hence, the total number of injections from set \(A\) to set \(B\) is 24. This corresponds to option (c) in the given choices.

Key Concepts

Injective Functions

An injective function is one where every element in the domain is mapped to a unique element in the codomain. This means no two elements from set \( A \) can share the same image in set \( B \). Imagine assignment of unique keys to lockers; each key opens only one locker.
In the context of our problem, given 3 elements in set \( A \) and 4 in set \( B \), we ensure each key (element from \( A \)) fits only one locker (unique in \( B \)). This characteristic is crucial for injections.
Understanding injective functions is important because it lays the foundation for bijections and surjections—other types of function mappings.

Map From One Set To Another

Mapping from one set to another means assigning each element of one set to one element of another set. In our scenario, elements of set \( A \) map to set \( B \).
For injective mapping, each item in the smaller set \( A \) must find its counterpart in set \( B \) such that no overlaps happen. Here’s a step-by-step visualization:

• Select one element in set \( A \), map it to any of the 4 possible in set \( B \).
• Choose the second element in set \( A \), this time from 3 remaining options in set \( B \).
• Finally, map the last element in \( A \) to one of the 2 left in set \( B \). This process represents an order-specific selection important in injections.

Combinatorial Functions

Combinatorial functions deal with combinations or arrangements of given items under specific conditions. In our injection problem, we're essentially asking how many different combinations can map from set \( A \) to set \( B \).
Using principles of combinatorics, this involves multiplying the number of choices available at each stage of selection:

• For the first choice: 4 possibilities
• For the second choice: 3 possibilities remain
• For the third choice: Only 2 choices are left

The calculation \(4 \times 3 \times 2\) falls under the category of permutations, a fundamental concept in combinatorics, allowing us to solve the problem by structured counting.

Function Mappings

Function mappings involve the process of assigning each element of a domain to elements in a codomain. Specifics of these mappings can vary significantly. In injections, mappings are one-to-one, creating a distinct connection between domain and codomain.
In our problem, mapping functions distinctly means taking into consideration each unique correspondence. Setting up mappings properly reveals how many ways our set \( A \), with 3 elements, fits into \( B \) with 4 spots without redundancy.
Real-life examples could include assigning seats in a theater or passwords to systems where repetition is not allowed. By understanding function mappings, students can delve deeper into complex algebraic structures and their applications.