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Problem 1 Part A Recall that we write x = y (mod p) for integers x, y, and p if there exists an integer c such that (x - y) = cp. For example, 16 = 2 (mod 7) because (16 - 2) = 2 × 7. Write a complete, careful proof that, for any p, the relation = (mod p) is an equivalence relation.

Problem 1 Part A Recall that we write x = y (mod p) for integers x, y, and p if there exists an integer c such that (x - y) = cp. For example, 16 = 2 (mod 7) because (16 - 2) = 2 × 7. Write a complete, careful proof that, for any p, the relation = (mod p) is an equivalence relation.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please help me solve these problems about relations

**Problem 1** **Part A** Recall that we write \( x \equiv y \pmod{p} \) for integers \( x, y, \) and \( p \) if there exists an integer \( c \) such that \( (x - y) = cp \). For example, \( 16 \equiv 2 \pmod{7} \) because \( (16 - 2) = 2 \times 7 \). Write a complete, careful proof that, for any \( p \), the relation \( \equiv \pmod{p} \) is an equivalence relation. **Part B** List all the equivalence classes of the relation \( \equiv \pmod{3} \). You can use either set-builder or roster notation. **Problem 2** Let \( A = \{ a, b, c, d, e \} \). Suppose that \( R \) is an equivalence relation on \( A \). Suppose further that \( R \) has two equivalence classes, and that \( aRd, bRc, \) and \( eRd \). Fully describe \( R \) by either writing it as a set or drawing it.
Transcribed Image Text:**Problem 1** **Part A** Recall that we write \( x \equiv y \pmod{p} \) for integers \( x, y, \) and \( p \) if there exists an integer \( c \) such that \( (x - y) = cp \). For example, \( 16 \equiv 2 \pmod{7} \) because \( (16 - 2) = 2 \times 7 \). Write a complete, careful proof that, for any \( p \), the relation \( \equiv \pmod{p} \) is an equivalence relation. **Part B** List all the equivalence classes of the relation \( \equiv \pmod{3} \). You can use either set-builder or roster notation. **Problem 2** Let \( A = \{ a, b, c, d, e \} \). Suppose that \( R \) is an equivalence relation on \( A \). Suppose further that \( R \) has two equivalence classes, and that \( aRd, bRc, \) and \( eRd \). Fully describe \( R \) by either writing it as a set or drawing it.
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The objective of the question is to check the given relation is an equivalence  relation. 

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