Problem 7 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(A=\\{1,2,3,4\\}\) and \(B=\\{1,2\\}\). Then, the number of onto functions from \(A\) to \(B\) is: (A) 8 (B) 14 (C) 12 (D) None of these

Step-by-Step Solution

Answer

The number of onto functions is 14, option (B).

Step 1: Understanding Onto Functions

An onto function, or surjective function, from a set \( A \) to a set \( B \) requires that every element in \( B \) is mapped to by at least one element in \( A \). In this problem, we have \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 2\} \).

Step 2: Total Possible Functions

The total number of functions from set \( A \) to set \( B \) is computed as \( |B|^{|A|}\). In this case, we have two elements in \( B \) and four in \( A \), so the total number of functions is \( 2^4 = 16 \).

Step 3: Calculating Non-Onto Functions

Non-onto functions occur when at least one element of \( B \) is not mapped to by any element of \( A \). If one particular element of \( B \) is not reached, all elements of \( A \) must be mapped to the other element in \( B \). There are 2 such non-onto functions: all elements map to 1 or all map to 2.

Step 4: Finding the Number of Onto Functions

Now, we subtract the number of non-onto functions from the total number of functions. Thus, the number of onto functions is \( 16 - 2 = 14 \).

Step 5: Final Answer

The number of onto functions from \( A \) to \( B \) is 14, which corresponds to option (B).

Key Concepts

Surjective Functions

Surjective functions, also known as "onto functions," are a fundamental aspect of function analysis in mathematics. To understand what a surjective function truly is, we must look at its action: every element in the target set, also known as the codomain, must have at least one element from the source set, or domain, mapping to it. This ensures that the entire range is covered.

When dealing with surjective functions, ask yourself:

• Is each element in the codomain being utilized?
• Does every element in the range correspond to at least one element in the domain?

For example, in our exercise, we have a function from set \( A \) to set \( B \). A function is considered surjective if both 1 and 2 in set \( B \) are mapped to by elements from set \( A \).

If you can show that no element is left out in \( B \), then the function from \( A \) to \( B \) is surjective.

Set Theory

Set theory is a branch of mathematical logic that investigates collections of objects, known as sets. It forms the foundation for various mathematical concepts including functions, mappings, and relations.

In our specific problem, we deal with two sets:

• Set \( A = \{1, 2, 3, 4\} \)
• Set \( B = \{1, 2\} \)

Set theory helps us understand how functions like the ones in our exercise operate between two sets. We aim to explore how elements from set \( A \) can be related to elements in set \( B \), through functions.

Knowing the number of elements in each set is crucial for determining the number of possible functions. For instance, the number of all possible functions from \( A \) to \( B \) is determined using the formula \( |B|^{|A|} \). Understanding sets, and their sizes, is the key starting point for exploring more about functions between these sets.

Function Mapping

Function mapping is an essential concept that explains how each element from one set is associated with an element from another set. In mathematical terms, it is often shown as \( f: A \to B \), meaning that \( f \) maps elements of set \( A \) into set \( B \).

Within the context of our exercise, mapping can either result in a surjective function or not. The exercise required us to count the number of onto or surjective mappings from \( A \) to \( B \).

Here's what we did:

• Found the total number of possible mappings: \( 2^4 = 16 \), since each element in \( A \) could map to any of the 2 elements in \( B \).
• Identified non-surjective mappings where either all values map to 1 or all to 2. This gave us 2 non-surjective functions.
• Subtracted the number of non-surjective functions from the total, giving us 14 onto functions.

Function mapping is all about understanding how elements connect between two sets. When practiced well, it becomes a valuable tool in deciphering complex mathematical concepts.