Problem 7
Question
Let \(E=\\{1,2,3,4\\}\) and \(F=\\{1,2\\}\), then the number of onto functions from \(E\) to \(F\) is [IIT Screening-2001] (a) 14 (b) 16 (c) 12 (d) 8
Step-by-Step Solution
Verified Answer
There are 14 onto functions from \(E\) to \(F\).
1Step 1: Understand the problem
We need to find the number of onto functions from set \(E\) to set \(F\). An onto function, also known as a surjective function, is one where every element in the codomain (set \(F\)) is mapped by at least one element from the domain (set \(E\)).
2Step 2: Total possible functions
Calculate the total number of functions from set \(E\) to set \(F\). Since \(E\) has 4 elements and \(F\) has 2 elements, each element in \(E\) can be mapped to any of the 2 elements in \(F\). Thus, there are \(2^4 = 16\) total functions.
3Step 3: Calculate functions that are not onto
Determine the number of functions that are not onto. A function is not onto if all elements of \(E\) map to the same single element of \(F\). There are 2 such functions: either all elements map to the first element of \(F\), or all map to the second element of \(F\).
4Step 4: Calculate onto functions
Subtract the number of non-onto functions from the total number of functions to find onto functions. Thus, the number of onto functions is \(16 - 2 = 14\).
Key Concepts
Surjective FunctionSet TheoryFunction MappingIIT JEE Mathematics
Surjective Function
A surjective function, also called an onto function, is a type of mapping in set theory. It ensures that every element in the codomain is the image of at least one element from the domain. In simpler terms, you can think of a surjective function as a delivery service where every unique address (an element in the codomain) must receive a package (an image from the domain).
For a function to be surjective, two key aspects need to be met:
For a function to be surjective, two key aspects need to be met:
- Each element in the codomain must be mapped from the domain.
- This means there cannot be any unused elements in the codomain.
Set Theory
Set theory is a branch of mathematical logic that studies collections of objects, known as sets. Understanding sets is fundamental because they form the basis for various mathematical concepts, including functions, relations, and numbers.
In set theory, a function between two sets, say \(E\) and \(F\), is viewed as a set of ordered pairs such that each element of \(E\) appears exactly once. Set \(E\) is called the domain, and \(F\) is the codomain.
In the exercise, set \(E = \{1,2,3,4\}\) is the domain of the function and is being mapped onto set \(F = \{1,2\}\), the codomain. Understanding the structure and properties of these sets allows you to apply the principles of surjectivity correctly.
In set theory, a function between two sets, say \(E\) and \(F\), is viewed as a set of ordered pairs such that each element of \(E\) appears exactly once. Set \(E\) is called the domain, and \(F\) is the codomain.
In the exercise, set \(E = \{1,2,3,4\}\) is the domain of the function and is being mapped onto set \(F = \{1,2\}\), the codomain. Understanding the structure and properties of these sets allows you to apply the principles of surjectivity correctly.
Function Mapping
Function mapping refers to the process of linking elements from one set (the domain) to elements in another set (the codomain). This is foundational in determining different types of functions, such as one-to-one, onto, and bijective functions.
Each element in the domain must be paired with an element in the codomain, but for the function to be considered onto, as required by the exercise, every element in the codomain needs to be associated with at least one element from the domain. In this exercise, finding the onto function involves:
Each element in the domain must be paired with an element in the codomain, but for the function to be considered onto, as required by the exercise, every element in the codomain needs to be associated with at least one element from the domain. In this exercise, finding the onto function involves:
- Calculating total possible mappings from the domain \(E\) to codomain \(F\).
- Determining mappings that do not cover all elements in the codomain.
IIT JEE Mathematics
IIT JEE Mathematics is a critical subject in the entrance examination for India’s prestigious engineering institutes, like the Indian Institutes of Technology (IITs). It tests students on their understanding and application of various advanced mathematical concepts, including functions, calculus, and algebra.
Questions on functions, like the one provided in the exercise, are common in IIT JEE math exams. Tackling such problems requires clear concepts and the ability to apply theoretical understanding effectively. Mastery of topics like surjective functions not only aids in solving these problems but also lays a strong foundation for more complex mathematical challenges faced during higher education.
Questions on functions, like the one provided in the exercise, are common in IIT JEE math exams. Tackling such problems requires clear concepts and the ability to apply theoretical understanding effectively. Mastery of topics like surjective functions not only aids in solving these problems but also lays a strong foundation for more complex mathematical challenges faced during higher education.
Other exercises in this chapter
Problem 5
\(f: R \rightarrow R, f(x)=(x-1)(x-2)(x-3)\) is [Roorkee-1999] (a) one-one but not onto (b) onto but not one-one (c) both one-one and onto (d) neither one-one n
View solution Problem 6
If \(f:[0, \infty) \rightarrow[0, \infty)\) and \(f(x)=\frac{x}{1+x}\), then (a) one-one and onto (b) one-one but not onto (c) onto but not one-one (d) neither
View solution Problem 8
Let \(f: R \rightarrow R\) be defined as \(f(x)=x|x| .\) Which one of the following is correct? (a) \(f\) is only onto (b) \(f\) is only one-one (c) \(f\) is ne
View solution Problem 9
Let \(f: N \rightarrow N\) defined by \(f(x)=x^{2}+x+1, x \in\) \(N\), then \(f\) is [AMU-2000] (a) one-one onto (b) Many one onto (c) one-one but not onto (d)
View solution