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Let = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence class containing a; that is, [a] = { c | cE A Ac = a }. Show that for every a and b in A, we have [a] = [b] if and only if a = b. [Hints. Proving an iff statement typically requires two separate proof steps, one for each implication direction. [a] and [b] are sets, so [a] = [b] means that Vc (c e [a] + c € [b]). By definition, an element c is in [a] if and only if c = a. In particular, a e [a].]

Let = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence class containing a; that is, [a] = { c | cE A Ac = a }. Show that for every a and b in A, we have [a] = [b] if and only if a = b. [Hints. Proving an iff statement typically requires two separate proof steps, one for each implication direction. [a] and [b] are sets, so [a] = [b] means that Vc (c e [a] + c € [b]). By definition, an element c is in [a] if and only if c = a. In particular, a e [a].]

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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Let = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence class containing a; that is, [a] = { c | c E A AC = a }. Show that for every a and b in A, we have [a] = [b] if and only if a = b. [Hints. Proving an iff statement typically requires two separate proof steps, one for each implication direction. [a] and [b] are sets, so [a] = [b] means that Vc (c e [a] → c € (b). By definition, an element c is in [a] if and only if c = a. In particular, a E [a].]
Transcribed Image Text:Let = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence class containing a; that is, [a] = { c | c E A AC = a }. Show that for every a and b in A, we have [a] = [b] if and only if a = b. [Hints. Proving an iff statement typically requires two separate proof steps, one for each implication direction. [a] and [b] are sets, so [a] = [b] means that Vc (c e [a] → c € (b). By definition, an element c is in [a] if and only if c = a. In particular, a E [a].]
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