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].]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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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].]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5655175-49d2-4db6-a7fd-785d70a02cda%2F0c835213-c62e-4591-b3d5-c29c65969e24%2Frc1ql07_processed.jpeg&w=3840&q=75)
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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