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Chapter 6: Problem 350

Another relation that you may have learned about in school, perhaps in the guise of "clock arithmetic," is the relation of equivalence modulo \(n\). For integers (positive, negative, or zero) \(a\) and \(b\), we write \(a \equiv b\) \((\bmod n)\) to mean that \(a-b\) is an integer multiple of \(n,\) and in this case, we say that \(a\) is congruent to \(b\) modulo \(n\) and write \(a \equiv b(\bmod n) . .\) Show that the relation of congruence modulo \(n\) is an equivalence relation.

Short Answer

Expert verified
The relation of congruence modulo n is reflexive, symmetric, and transitive.

Step by step solution

01

- Define an equivalence relation

An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity. We need to show that congruence modulo n satisfies all three.
02

- Show reflexivity

For any integer a, we need to show that a 鈮� a (mod n). Since a - a = 0 and 0 is an integer multiple of n (0*n), it follows that a 鈮� a (mod n).
03

- Show symmetry

For any integers a and b, if a 鈮� b (mod n), then we need to show b 鈮� a (mod n). Given a 鈮� b (mod n), it means that a - b is a multiple of n. Thus, b - a = - (a - b). As - (a - b) is also a multiple of n, it follows that b 鈮� a (mod n).
04

- Show transitivity

For any integers a, b, and c, if a 鈮� b (mod n) and b 鈮� c (mod n), then we need to show that a 鈮� c (mod n). Given a 鈮� b (mod n) implies a - b is a multiple of n, and b 鈮� c (mod n) implies b - c is a multiple of n. Adding these two congruences, we get (a - b) + (b - c) = a - c, which is also a multiple of n. Therefore, a 鈮� c (mod n).

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Key Concepts

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

Modular Arithmetic
Modular arithmetic, often referred to as 'clock arithmetic,' is a system of arithmetic for integers. It considers numbers to wrap around after reaching a certain value, called the modulus. For example, if we use a modulus of 12, the numbers reset after 12. So, 13 mod 12 would be 1. This approach simplifies calculations and finds practical applications in fields like cryptography.
Congruence Modulo n
In modular arithmetic, two integers are said to be congruent modulo n if they give the same remainder when divided by n. We write this as \(a \equiv b (\bmod n)\). Essentially, this means that \(a - b\) is an integer multiple of n. For example, if \(a = 17\) and \(b = 5\), we can say \(17 \equiv 5 (\bmod 12)\) because 17 and 5 differ by 12, a multiple of 12.
Equivalence Properties
For any relation to be classified as an equivalence relation, it must satisfy three fundamental properties: reflexivity, symmetry, and transitivity. These properties ensure that the relation can be consistently applied across various pairs of elements in the set, maintaining a structured and predictable behavior.
Reflexivity
Reflexivity is the property where every element is related to itself. In the context of congruence modulo n, this means that any integer a is congruent to itself modulo n. To see why, consider that a - a = 0, and 0 is indeed a multiple of any number n (since 0 = 0*n). Therefore, \(a \equiv a (\bmod n)\) holds true for any integer a.
Symmetry
Symmetry implies that if one element is related to another, the relation works in the reverse direction as well. For congruence modulo n, if \(a \equiv b (\bmod n)\), then \(b \equiv a (\bmod n)\). This is because if \(a - b\) is a multiple of n, then it follows that \(b - a\) (which is simply the negative of \(a - b\)) is also a multiple of n.
Transitivity
Transitivity requires that if one element is related to a second, and that second element is related to a third, then the first element must also be related to the third. For congruence modulo n, this means if \(a \equiv b (\bmod n)\) and \(b \equiv c (\bmod n)\), then \(a \equiv c (\bmod n)\). This holds true because if \(a - b\) is a multiple of n and \(b - c\) is a multiple of n, then adding these results, we find that \(a - c\) must also be a multiple of n.

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Most popular questions from this chapter

Here are some questions that will help you get used to the formal idea of a relation and the related formal idea of a function. \(S\) will stand for a set of size \(s\) and \(T\) will stand for a set of size \(t\). (a) What is the size of the largest relation from \(S\) to \(T ?\) (b) What is the size of the smallest relation from \(S\) to \(T ?\) (c) The relation of a function \(f: S \rightarrow T\) is the set of all ordered pairs \((x, f(x))\) with \(x \in S .\) What is the size of the relation of a function from \(S\) to \(T ?\) That is, how many ordered pairs are in the relation of a function from \(S\) to \(T ?\) (h) (d) We say \(f\) is a one-to-one function or injection from \(S\) to \(T\) if each member of \(S\) is related to a different element of \(T\). How many different elements must appear as second elements of the ordered pairs in the relation of a one-to-one function (injection) from \(S\) to \(T ?\) (e) A function \(f: S \rightarrow T\) is called an onto function or surjection if each element of \(T\) is \(f(x)\) for some \(x \in S\) What is the minimum size that \(S\) can have if there is a surjection from \(S\) to \(T ?\)

Explain why a bijection must have an inverse.

Explain why a function that has an inverse must be a bijection.

Let \(\sigma\) and \(\varphi\) be permutations. (a) Why must \(\sigma \circ \varphi\) have an inverse? (b) Is \((\sigma \circ \varphi)^{-1}=\sigma^{-1} \varphi^{-1}\) ? (Prove or give a counter-example.) (h) (c) Is \((\sigma \circ \varphi)^{-1}=\varphi^{-1} \sigma^{-1} ?\) (Prove or give a counter-example.)

How can you compute the Orbit Enumerator of G acting on functions from \(S\) to a finite set \(T\) from the cycle index of Gacting on \(S\) ? (Use \(t\), thought of as a variable, as the picture of an element \(t\) of \(T .)\) State and prove the relevant theorem! This is P贸lya's and Redfield's famous enumeration theorem.
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