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roblem 0.2. Which of the following examples are equivalence relations? (1) S is the set of real numbers and I = y if |x – y| < 1 (2) S is the set of integers and a = b if a – b is divisible by 3

roblem 0.2. Which of the following examples are equivalence relations? (1) S is the set of real numbers and I = y if |x – y| < 1 (2) S is the set of integers and a = b if a – b is divisible by 3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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As we have seen with fractions, there is often more than one notions of "sameness" that is relevant to given problem. Mathematicians have abstracted the notion to something called an equivalence relation. Given a set S of objects, an equivalence relation on S is a way of dividing S up into subsets of elements that are "the same" in some sense. When we want to say that a and b are the same from our perspective, we write a = b. This is clearly similar to the symbol for equality but different enough to remind us that we are not saying a and b are equal. For fractions, our set was {(n, d) : n e Z, d e N} where we think of n and d as being the numerator and denominator. Mostly we do not care about the specifics of these two numbers, only the value of the ratio, so we consider (1, 2) and (3, 6) equivalent even though they are not identical. In general we decided that (a, b) = (c, d) if ad = bc. For this to make sense as a kind of "equal", the following things need to be checked: SO Problem 0.1. Let S = {(n, d) : n E Z, de N} and define (a, b) = (c, d) to mean ad (1) For any (n, d) we have (n, d) = (n, d) (2) If (a, b) = (c, d) then (c, d) = (a, b) (3) If (a, b) = (c, d) and (c, d) = (e, f) then (a, b) = (e, f) = bc These three properties are the axioms for something to be considered an equivalence relation. Problem 0.2. Which of the following examples are equivalence relations? (1) S is the set of real numbers and r = y if |x – y| < 1 (2) S is the set of integers and a = b if a – b is divisible by 3 (3) S is the set of natural numbers and a = b is a < b (4) S is the set of real numbers and x = y if sin r = sin y (5) S is the set of English words, and x = y if x and y are synonyms (6) S is the set of English words, and r = y if x and y are antonyms (7) S is the set of English words, and x = y if x and y start with the same letter (8) S is the set of English words, and x = y if x and y have the same number of syllables
Transcribed Image Text:As we have seen with fractions, there is often more than one notions of "sameness" that is relevant to given problem. Mathematicians have abstracted the notion to something called an equivalence relation. Given a set S of objects, an equivalence relation on S is a way of dividing S up into subsets of elements that are "the same" in some sense. When we want to say that a and b are the same from our perspective, we write a = b. This is clearly similar to the symbol for equality but different enough to remind us that we are not saying a and b are equal. For fractions, our set was {(n, d) : n e Z, d e N} where we think of n and d as being the numerator and denominator. Mostly we do not care about the specifics of these two numbers, only the value of the ratio, so we consider (1, 2) and (3, 6) equivalent even though they are not identical. In general we decided that (a, b) = (c, d) if ad = bc. For this to make sense as a kind of "equal", the following things need to be checked: SO Problem 0.1. Let S = {(n, d) : n E Z, de N} and define (a, b) = (c, d) to mean ad (1) For any (n, d) we have (n, d) = (n, d) (2) If (a, b) = (c, d) then (c, d) = (a, b) (3) If (a, b) = (c, d) and (c, d) = (e, f) then (a, b) = (e, f) = bc These three properties are the axioms for something to be considered an equivalence relation. Problem 0.2. Which of the following examples are equivalence relations? (1) S is the set of real numbers and r = y if |x – y| < 1 (2) S is the set of integers and a = b if a – b is divisible by 3 (3) S is the set of natural numbers and a = b is a < b (4) S is the set of real numbers and x = y if sin r = sin y (5) S is the set of English words, and x = y if x and y are synonyms (6) S is the set of English words, and r = y if x and y are antonyms (7) S is the set of English words, and x = y if x and y start with the same letter (8) S is the set of English words, and x = y if x and y have the same number of syllables
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