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Chapter 6: Problem 12

Let \(A\) be a nonempty set. The identity function on the set \(A,\) denoted by \(I_{A},\) is the function \(I_{A}: A \rightarrow A\) defined by \(I_{A}(x)=x\) for every \(x\) in \(A\). Is \(I_{A}\) an injection? Is \(I_{A}\) a surjection? Justify your conclusions.

Short Answer

Expert verified
The identity function \(I_A\) on a nonempty set \(A\) is both an injection and a surjection because for any \(x_1, x_2 \in A\), \(I_A(x_1) = I_A(x_2)\) implies \(x_1 = x_2\), and for every element \(y \in A\), there exists an element \(x \in A\) such that \(I_A(x) = y\).

Step by step solution

01

Verify if \(I_A\) is an injection

To verify if \(I_A\) is an injection, we need to check if for all \(x_1, x_2 \in A\), \(I_A(x_1) = I_A(x_2)\) implies \(x_1 = x_2\). By the definition of the identity function, we have: \(I_A(x_1) = x_1\) and \(I_A(x_2) = x_2\) Now, if \(I_A(x_1) = I_A(x_2)\), then \(x_1 = x_2\). This is true for all \(x_1, x_2 \in A\). Thus, \(I_A\) is an injection.
02

Verify if \(I_A\) is a surjection

To verify if \(I_A\) is a surjection, we need to check if for every element \(y \in A\), there exists an element \(x \in A\) such that \(I_A(x) = y\). By the definition of the identity function, we have: \(I_A(x) = x\) Now, for any \(y \in A\), we can find an element \(x = y\) in \(A\) such that \(I_A(x) = x = y\). Thus, \(I_A\) is a surjection.
03

Conclusion

The identity function \(I_A\) is both an injection and a surjection, as it meets the criteria for both definitions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Injection
An injection, also called a one-to-one function, is a function where each element of the domain is mapped to a unique element in the co-domain, meaning no two distinct elements in the domain map to the same element in the co-domain. Think of it like assigning each student a unique locker; no two students end up with the same locker.

For the identity function, denoted as \(I_A\), each element \(x\) in the set \(A\) maps to itself. To determine if \(I_A\) is an injection, we need to check if \(I_A(x_1) = I_A(x_2)\) implies that \(x_1 = x_2\). Since \(I_A(x) = x\) by definition, it is clear that if \(x_1\) and \(x_2\) produce the same output, they must be the same element. Therefore, the identity function \(I_A\) is indeed an injection.
Surjection
Surjection, also known as an onto function, ensures that every element of the co-domain is an image of at least one element from the domain. Imagine having a cookie for each guest at a party, ensuring that each guest has one cookie at the end of the night.

For the identity function \(I_A\), each element \(x\) in the set \(A\) is mapped to itself. We determine if \(I_A\) is a surjection by checking if for every element \(y\) in the co-domain \(A\), there exists an element \(x\) in \(A\) such that \(I_A(x) = y\). Since \(I_A(x) = x\), we can always pick \(x = y\). Thus, every element in \(A\) is mapped from some element in \(A\), confirming that \(I_A\) is a surjection.
Set Theory
Set theory is the mathematical study of sets, which are collections of objects. It forms the basis of several branches of mathematics. In set theory, functions are understood through their behavior in terms of mappings and relationships between sets.

In discussing functions like the identity function or characteristics like injection and surjection, set theory provides the framework. The set \(A\) and the mapping \(I_A: A \rightarrow A\) exhibit the richness of set theory by illustrating relationships within a set. An identity function itself is an essential construct in set theory, illustrating how elements relate to themselves within the same set.

Understanding set theory aids in comprehending how functions like \(I_A\) perform and highlights their properties as both injective and surjective. Every element being mapped to itself in the identity function is a simple yet powerful demonstration of the principles of set theory.

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Most popular questions from this chapter

Let \(R_{6}=\\{0,1,2,3,4,5\\} .\) Define \(f: R_{6} \rightarrow R_{6}\) by \(f(x)=x^{2}+4(\bmod 6),\) and define \(g: R_{6} \rightarrow R_{6}\) by \(g(x)=(x+1)(x+4)(\bmod 6)\). (a) Calculate \(f(0), f(1), f(2), f(3), f(4),\) and \(f(5)\). (b) Calculate \(g(0), g(1), g(2), g(3), g(4),\) and \(g(5)\). (c) Is the function \(f\) equal to the function \(g ?\) Explain.

The number of divisors function. Let \(d\) be the function that associates with each natural number the number of its natural number divisors. That is, \(d: \mathbb{N} \rightarrow \mathbb{N}\) where \(d(n)\) is the number of natural number divisors of \(n\). For example, \(d(6)=4\) since \(1,2,3,\) and 6 are the natural number divisors of 6 (a) Calculate \(d(k)\) for each natural number \(k\) from 1 through 12 . (b) Does there exist a natural number \(n\) such that \(d(n)=1 ?\) What is the set of preimages of the natural number \(1 ?\) (c) Does there exist a natural number \(n\) such that \(d(n)=2 ?\) If so, determine the set of all preimages of the natural number \(2 .\) (d) Is the following statement true or false? Justify your conclusion. For all \(m, n \in \mathbb{N},\) if \(m \neq n,\) then \(d(m) \neq d(n) .\) (e) Calculate \(d\left(2^{k}\right)\) for \(k=0\) and for each natural number \(k\) from 1 through 6 (f) Based on your work in Exercise (6e), make a conjecture for a formula for \(d\left(2^{n}\right)\) where \(n\) is a nonnegative integer. Then explain why your conjecture is correct. (g) Is the following statement is true or false? For each \(n \in \mathbb{N},\) there exists a natural number \(m\) such that \(d(m)=n\)

Prove Part (2) of Corollary 6.28. Let \(A\) and \(B\) be nonempty sets and let \(f: A \rightarrow B\) be a bijection. Then for every \(y\) in \(B,\left(f \circ f^{-1}\right)(y)=y\).

Functions Whose Domain is \(\mathcal{M}_{2}(\mathbb{R}) .\) Let \(\mathcal{M}_{2}(\mathbb{R})\) represent the set of all 2 by 2 matrices over \(\mathbb{R}\). (a) Define det: \(\mathcal{M}_{2}(\mathbb{R}) \rightarrow \mathbb{R}\) by $$\operatorname{det}\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=a d-b c$$ This is the determinant function introduced in Exercise (9) from Section \(6.2 .\) Is the determinant function an injection? Is the determinant function a surjection? Justify your conclusions. (b) Define tran: \(\mathcal{M}_{2}(\mathbb{R}) \rightarrow \mathcal{M}_{2}(\mathbb{R})\) by $$\operatorname{tran}\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=A^{T}=\left[\begin{array}{ll} a & c \\ b & d \end{array}\right]$$ This is the transpose function introduced in Exercise (10) from Section 6.2 . Is the transpose function an injection? Is the transpose function a surjection? Justify your conclusions. (c) Define \(F: \mathcal{M}_{2}(\mathbb{R}) \rightarrow \mathbb{R}\) by $$F\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=a^{2}+d^{2}-b^{2}-c^{2}$$ Is the function \(F\) an injection? Is the function \(F\) a surjection? Justify your conclusions.

Let \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) be defined by \(f(m)=2 m+1\). (a) Evaluate \(f(-7), f(-3), f(3),\) and \(f(7)\). * (b) Determine the set of all of the preimages of 5 and the set of all of the preimages of \(4 .\) (c) Determine the range of the function \(f\). (d) Sketch a graph of the function \(f\). See the comments in Exercise (3d).
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