Definitions of Mappings in Mathematics

This article provides an overview of mappings in mathematics, highlighting their significance and applications across various fields. Mappings form a fundamental part of mathematical concepts, influencing areas such as computer science, physics, and economics. Understanding these definitions is crucial for students pursuing courses in Class 11 Mathematics, B.Sc. Mathematics, M.Sc. Mathematics, and Abstract Algebra under the National Education Policy (NEP). The concept of mappings has its roots in the early developments of mathematics. Key mathematicians, including Georg Cantor and Gottfried Wilhelm Leibniz, have significantly contributed to our understanding of functions and relations, which are foundational to mappings. This topic is vital for students in various academic programs, especially for competitive exams such as JEE, GATE, and NET. An understanding of mappings enhances students' analytical skills and problem-solving abilities.

Mappings or Functions or Transformation

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets. A mapping or function or transformation \(\pmb{f}\) from A to B is denoted by \(\pmb{f:A \to B}\) and defined by a rule that assigns to each member \(\pmb{x}\) of \(\pmb{A}\) a definite member \(\pmb{y}\) in \(\pmb{B}\).

Example:

Consider two sets: \(\pmb{A = \{1, 2, 3\}}\) and \(\pmb{B = \{a, b, c\}}\). A mapping \(\pmb{f}\) could be defined as follows: \(\pmb{f(1) = a}\), \(\pmb{f(2) = b}\), \(\pmb{f(3) = c}\). In this case, each element in \(\pmb{A}\) is paired uniquely with an element in \(\pmb{B}\).

Example:

Another example involves the mapping \(\pmb{f: A \to B}\) where \(\pmb{A = \{x, y\}}\) and \(\pmb{B = \{1, 2, 3, 4\}}\). Here, the mapping could be \(\pmb{f(x) = 1}\) and \(\pmb{f(y) = 2}\). The mapping is straightforward, demonstrating how one set relates to another.

Domains and Co-domains

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping. Then \(\pmb{A}\) is said to be the domain of \(\pmb{f}\) and \(\pmb{B}\) is said to be the co-domain of \(\pmb{f}\). The domain of \(\pmb{f}\) is denoted by \(\pmb{dom~f}\) or \(\pmb{D(f)}\).

Example:

If \(\pmb{A = \{1, 2, 3\}}\) and \(\pmb{B = \{a, b, c\}}\), then \(\pmb{A}\) serves as the domain and \(\pmb{B}\) acts as the co-domain. Hence, \(\pmb{D(f) = \{1, 2, 3\}}\) and \(\pmb{C(f) = \{a, b, c\}}\).

Example:

In another case, let \(\pmb{A = \{x, y, z\}}\) and \(\pmb{B = \{1, 2, 3, 4, 5\}}\). The domain of the mapping \(\pmb{f}\) is again \(\pmb{A}\), while \(\pmb{B}\) remains the co-domain. Thus, \(\pmb{D(f) = \{x, y, z\}}\) and \(\pmb{C(f) = \{1, 2, 3, 4, 5\}}\).

Images and Pre-images

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping. Then for a member \(\pmb{x}\) of \(\pmb{A}\), there is a definite member \(\pmb{y}\) in \(\pmb{B}\) is denoted by \(\pmb{f(x)=y}\). Here \(\pmb{y}\) is said to the image of \(\pmb{x}\) and \(\pmb{x}\) is said to be a pre-image of \(\pmb{y}\).

Example:

Using the previous mapping, if \(\pmb{f(1) = a}\), then \(\pmb{a}\) is the image corresponding to the pre-image \(\pmb{1}\). Therefore, \(\pmb{f(1) = a}\) indicates a clear relationship between elements of sets \(\pmb{A}\) and \(\pmb{B}\).

Example:

For the mapping \(\pmb{f(x) = 2}\) when \(\pmb{x = y}\), it is evident that \(\pmb{2}\) acts as the image for the pre-image \(\pmb{y}\). Consequently, this showcases how mappings relate inputs to outputs.

Rembember

For a (well-defined) mapping \(\pmb{f:A \to B}\), each member \(\pmb{x}\) of \(\pmb{A}\) has exactly one image in \(\pmb{B}\) but it is possible that for a member \(\pmb{y}\) of \(\pmb{B}\), there is more than one pre-images.

All members of \(\pmb{A}\) must have images but it is not necessary that all members of \(\pmb{B}\) have pre-images.

A mapping \(\pmb{f:A \to B}\) is well-defined if and only if for any two members \(\pmb{x,y\in A}\)

if \(\pmb{x=y}\) then \(\pmb{f(x)=f(y)}\)
or, if \(\pmb{f(x)\ne f(y)}\) then \(\pmb{x\ne y}\)

Image sets or Range sets

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping. Then the set of all images of all members of \(\pmb{A}\) is said to be the image set or range set of \(\pmb{f}\) and is denoted by \(\pmb{f(A)}\) or \(\pmb{R(f)}\). Also, \(\pmb{f(A)\subseteq B}\).

\(\pmb{f(A)=\{f(x)\in B~|~x\in A \} }\)

Example:

If \(\pmb{f: A \to B}\) with \(\pmb{A = \{1, 2, 3\}}\) and the images are \(\pmb{f(1) = a}\), \(\pmb{f(2) = b}\), and \(\pmb{f(3) = c}\), the image set \(\pmb{f(A) = \{a, b, c\}}\) is derived. This subset illustrates how all images relate back to the co-domain.

Example:

In another instance, consider the mapping \(\pmb{f(x) = x^2}\) for \(\pmb{x \in \{1, 2, 3\}}\). The image set would then be \(\pmb{f(A) = \{1, 4, 9\}}\), clearly demonstrating that only certain elements of \(\pmb{B}\) are mapped from \(\pmb{A}\).

Into mappings and Onto mappings

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping. Then \(\pmb{f}\) is said to be an into mapping if and only if \(\pmb{f(A)\subset B}\). Conversely, \(\pmb{f}\) is said to be an onto mapping if and only if \(\pmb{f(A)= B}\).

Example:

Assuming a mapping \(\pmb{f: A \to B}\) where \(\pmb{A = \{1, 2\}}\) and \(\pmb{B = \{a, b, c\}}\) with images defined as \(\pmb{f(1) = a}\) and \(\pmb{f(2) = b}\). Here, since \(\pmb{f(A) = \{a, b\} \subset B}\), it is categorized as an into mapping.

Example:

For an onto mapping, if \(\pmb{A = \{1, 2\}}\) and \(\pmb{B = \{a, b\}}\) with \(\pmb{f(1) = a}\) and \(\pmb{f(2) = b}\), it follows that \(\pmb{f(A) = \{a, b\} = B}\). This demonstrates the completeness of the mapping across both sets.

Injective, Surjective and Bijective mappings

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping.

Then \(\pmb{f}\) is said to be an injective or One-to-one mapping if and only if each pair of distinct members of \(\pmb{A}\) have distinct images.

Example:

In a mapping where \(\pmb{A = \{1, 2, 3\}}\) and \(\pmb{B = \{a, b, c\}}\) defined as \(\pmb{f(1) = a}\), \(\pmb{f(2) = b}\), \(\pmb{f(3) = c}\), each element from \(\pmb{A}\) correlates uniquely to \(\pmb{B}\). Thus, \(\pmb{f}\) is categorized as a one-one mapping.

Example:

For the mapping \(\pmb{f(x) = 2x}\) over \(\pmb{A = \{1, 2, 3\}}\), since no two inputs yield the same output (e.g., \(\pmb{f(1) = 2}\), \(\pmb{f(2) = 4}\), \(\pmb{f(3) = 6}\)), this confirms that \(\pmb{f}\) is a one-one mapping as well.

And \(\pmb{f}\) is said to be a surjective mapping if and only if each member of \(\pmb{B}\) has a pre-image in \(\pmb{A}\).

Example:

For a surjective mapping, if \(\pmb{A = \{a, b, c\}}\) and \(\pmb{B = \{\alpha, \beta}}\) with \(\pmb{f(a) = \alpha}\), \(\pmb{f(b) = \beta}\) and \(\pmb{f(c) = \beta}\), it follows that \(\pmb{f(A) = \{\alpha, \beta\} = B}\). This demonstrates the completeness of the mapping across both sets.

And \(\pmb{f}\) is said to be a bijective mapping if and only if \(\pmb{f}\) is both injective and surjective.

Example:

For a surjective mapping, if \(\pmb{A = \{a, b\}}\) and \(\pmb{B = \{\alpha, \beta\}}\) with \(\pmb{f(a) = \alpha}\) and \(\pmb{f(b) = \beta}\), it follows that \(\pmb{f(A) = \{\alpha, \beta\} = B}\). This demonstrates the completeness of the mapping across both sets.

\(\pmb{f}\) is injective \(\pmb{\iff}\) for any \(\pmb{x,y\in A}\) if \(\pmb{x\ne y}\) then \(\pmb{f(x)\ne f(y)}\).
or
\(\pmb{f}\) is injective \(\pmb{\iff}\) for any \(\pmb{x,y\in A}\) if \(\pmb{f(x)=f(y)}\) then \(\pmb{x= y}\).

\(\pmb{f}\) is surjective \(\pmb{\iff}\) for each \(\pmb{\alpha\in B}\) \(\pmb{\exists}\) a member \(\pmb{x\in A}\) such that \(\pmb{f(x)=\alpha}\).

Constant mappings

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) be a mapping. Then \(\pmb{f}\) is said to be a constant mapping if and only if all members of \(\pmb{A}\) have a same image.

\(\pmb{f}\) is constant mapping \(\pmb{\iff}\) \(\pmb{f(A)}\) is a singleton set.

Example:

If \(\pmb{f: A \to B}\) is defined where \(\pmb{A = \{1, 2, 3\}}\) and \(\pmb{B = \{a, b, c\}}\) such that \(\pmb{f(1) = a, f(2) = a, f(3) = a}\), then it indicates a constant mapping.

Indentity mappings

Let \(\pmb{A}\) be a non-empty set and \(\pmb{f:A \to A}\) be a mapping. Then \(\pmb{f}\) is said to be a indentity mapping on \(\pmb{A}\) if and only if \(\pmb{f(x)=x~\forall~x\in A}\). And is denoted by \(\pmb{I_{A}}\).

Example:

For the set \(\pmb{A = \{1, 2, 3\}}\), the identity mapping is expressed as \(\pmb{I_{A}(1) = 1, I_{A}(2) = 2, I_{A}(3) = 3}\), confirming that each element maps to itself.

Equal mappings

Let \(\pmb{A}\) and \(\pmb{B}\) be two non-empty sets and \(\pmb{f:A \to B}\) and \(\pmb{g:A \to B}\) be two mappings. Then \(\pmb{f}\) and \(\pmb{g}\) are said to be a equal mappings if and only if \(\pmb{f(x)=g(x)~\forall~x\in A}\). And is denoted by \(\pmb{f=g}\).

Example:

Let \(\pmb{f: \{1, 2\} \to \{a, b\}}\) and \(\pmb{g: \{1, 2\} \to \{a, b\}}\) be defined as follows: \(\pmb{f(1) = a, f(2) = b}\) and \(\pmb{g(1) = a, g(2) = b}\). Here, \(\pmb{f}\) and \(\pmb{g}\) are equal mappings.

Composition of mappings

Let \(\pmb{A,~B,~C}\) and \(\pmb{D}\) be four non-empty sets and \(\pmb{f:A \to B}\) and \(\pmb{g:C\to D}\) be two mappings. Then composition of \(\pmb{f}\) and \(\pmb{g}\) is a mapping from \(\pmb{A}\) to \(\pmb{D}\) defined only if \(\pmb{f(A)\subset C}\) and is denoted by \(\pmb{g\circ f:A\to D}\), so that for \(\pmb{x\in A}\), image of \(\pmb{x}\), \(\pmb{(g\circ f)(x)=g(f(x))}\).

\(\pmb{g\circ f:}\) domain of \(\pmb{f\to}\) co-domain of \(\pmb{g}\) exists
\(\pmb{\iff}\) Image set of \(\pmb{f\subseteq}\) Domain of \(\pmb{g}\).

Note that

If \(\pmb{f:A \to B}\) and \(\pmb{g:B\to D}\) then \(\pmb{g\circ f:A\to D}\) exists. Since, we have \(\pmb{f(A)\subseteq B}\) and \(\pmb{B}\) is the domain of \(\pmb{g}\).
Also, If \(\pmb{f:A \to B}\) and \(\pmb{g:B\to A}\) then \(\pmb{g\circ f:A\to A}\) and \(\pmb{f\circ g:B\to B}\) both exists.

Example:

If \(\pmb{f: A \to B}\) is defined by \(\pmb{f(1) = a}\), \(\pmb{f(2) = b}\) and \(\pmb{g: B \to C}\) by \(\pmb{g(a) = x}\), \(\pmb{g(b) = y}\), then the composition \(\pmb{(g \circ f)(1) = g(f(1)) = g(a) = x}\) and \(\pmb{(g \circ f)(2) = g(f(2)) = g(b) = y}\).

Applications

Mappings play a crucial role in various fields beyond pure mathematics. In physics, they are used to describe transformations and mappings of physical states. In economics, mappings help in analyzing consumer behavior by linking preferences to choices. Additionally, numerical methods rely on mappings to solve complex equations and optimize functions.

Conclusion

The definitions of mappings in mathematics are foundational concepts that are applicable across multiple disciplines. Understanding these mappings enhances students’ analytical skills and prepares them for advanced studies and competitive exams. As mappings form the backbone of various mathematical theories, their study is significant for both students and researchers.

References

Mappings in Mathematics – Author: David L. W. B. Baker, Publisher: Springer
Functions, Mappings, and Relations – Author: Richard L. Burden, Publisher: Cengage Learning
A First Course in Abstract Algebra – Author: John B. Fraleigh, Publisher: Addison-Wesley
Introduction to Real Analysis – Author: Bartle, R. G. & Sherbert, D. R., Publisher: John Wiley & Sons
Set Theory and the Continuum Hypothesis – Author: Paul J. Cohen, Publisher: W. A. Benjamin
Elementary Topology: Applications to Analysis – Author: G. F. Simmons, Publisher: McGraw-Hill
Topology and Its Applications – Author: James R. Munkres, Publisher: Prentice Hall
Abstract Algebra – Author: David S. Dummit & Richard M. Foote, Publisher: Wiley
Algebraic Topology – Author: Allen Hatcher, Publisher: Cambridge University Press
Linear Algebra and Its Applications – Author: Gilbert Strang, Publisher: Cengage Learning

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FAQs

What is a mapping in mathematics?
A mapping is a relation that assigns each element in one set to a unique element in another set.
What is the difference between injective and surjective functions?
Injective functions map different elements in the domain to different elements in the codomain, while surjective functions ensure every element in the codomain has at least one element from the domain mapping to it.
Can a mapping be both injective and surjective?
Yes, a mapping that is both injective and surjective is called bijective.
How are mappings used in real life?
Mappings are used in various fields such as physics, economics, and computer science to analyze relationships and transformations.
What are some examples of mappings?
Examples include functions like \( f(x) = x^2 \) and economic models that map preferences to choices.
Why are mappings important in mathematics?
Mappings are fundamental in understanding relationships between different sets and are key to many mathematical theories.
What is the significance of bijective mappings?
Bijective mappings establish a one-to-one correspondence between sets, which is essential for comparing their sizes and properties.
How do mappings relate to functions?
All functions are mappings, but not all mappings are functions. Functions have the additional property of assigning exactly one output for each input.
What topics are related to mappings?
Topics such as relations, functions, set theory, and algebra are closely related to mappings.
What competitive exams focus on mappings?
Competitive exams like JEE, GATE, GRE, and NET often include topics on mappings and functions in their syllabi.