11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a) For each a, b eE A, a × b if and only if [a] N [b] = Ø. (b) For each a, b e A, if [a] # [b], then [a] N [b] = Ø. (c) For each a, b e A, if [a] N [b] 7 Ø then [a] = [b].
11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a) For each a, b eE A, a × b if and only if [a] N [b] = Ø. (b) For each a, b e A, if [a] # [b], then [a] N [b] = Ø. (c) For each a, b e A, if [a] N [b] 7 Ø then [a] = [b].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Answer 11b
![11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove
each of the following:
(a) For each a, b e A, a × b if and only if [a] N [b] = Ø.
(b) For each a, b e A, if [a] # [b], then [a] N [b] = Ø.
(c) For each a, b e A, if [a] N [b] # Ø then [a] = [b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79fec644-cc9a-4e79-932f-9a07bdd10b3d%2F8169d164-28d0-45fc-bcd4-bc973e9398e4%2Fk5514t6_processed.png&w=3840&q=75)
Transcribed Image Text:11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove
each of the following:
(a) For each a, b e A, a × b if and only if [a] N [b] = Ø.
(b) For each a, b e A, if [a] # [b], then [a] N [b] = Ø.
(c) For each a, b e A, if [a] N [b] # Ø then [a] = [b].
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