what is equivalence relation and invertibe??
 

Asked by Ohi | 13th Jun, 2017, 12:06: PM

Expert Answer:

begin mathsize 16px style Equivalence space relation colon
straight A space relation space straight R space on space straight a space set space straight A space is space said space to space be space an space equivalence space relation space on space straight A space
if space and space only space if
1. space it space is space reflexive. space left parenthesis straight a comma straight a right parenthesis element of straight R space for space all space straight a element of straight A
2. space it space is space symmetric. space left parenthesis straight a comma straight b right parenthesis space element of straight R space rightwards double arrow left parenthesis straight b comma straight a right parenthesis element of straight R space for space all space straight a comma space straight b element of straight R
3. space it space is space transitive. space left parenthesis straight a comma straight b right parenthesis element of straight R space and space left parenthesis straight b comma straight c right parenthesis element of straight R space rightwards double arrow left parenthesis straight a comma straight c right parenthesis element of straight R space for space all space straight a comma straight b comma straight c element of straight R
for space example comma
straight A thin space equals space open curly brackets 1 comma 2 comma 3 close curly brackets space and space straight B equals open curly brackets 1 comma 2 close curly brackets
rightwards double arrow straight R open curly brackets left parenthesis 1 comma 1 right parenthesis comma left parenthesis 1 comma 2 right parenthesis comma left parenthesis 2 comma 1 right parenthesis comma left parenthesis 2 comma 2 right parenthesis comma left parenthesis 3 comma 1 right parenthesis comma left parenthesis 3 comma 2 right parenthesis close curly brackets
Here comma space left parenthesis 1 comma 1 right parenthesis comma left parenthesis 2 comma 2 right parenthesis element of straight R space rightwards double arrow we space can space say space that space function space is space reflexive.
Also comma space left parenthesis 1 comma 2 right parenthesis space and space left parenthesis 2 comma 1 right parenthesis space element of straight R
Consider comma space left parenthesis 3 comma 2 right parenthesis space and space left parenthesis 2 comma 1 right parenthesis space rightwards double arrow left parenthesis 3 comma 2 right parenthesis element of straight R space
Hence comma space it space is space transitive.
Equivalence space relation space on space straight R.


Invertible space relation colon
Let comma space straight A comma space straight B space be space two space sets space and space let space straight R space be space straight a space relation space from space straight a space set space straight A space to space straight a space set space straight B. space
Then space the space inverse space of space straight R comma space denoted space by space straight R to the power of negative 1 end exponent equals open curly brackets open parentheses straight b comma straight a close parentheses colon open parentheses straight a comma straight b close parentheses element of straight R close curly brackets
Clearly comma space open parentheses straight a comma straight b close parentheses space element of straight R space left right double arrow open parentheses straight b comma straight a close parentheses element of straight R to the power of negative 1 end exponent
Also comma space Domain space left parenthesis straight R right parenthesis space equals space Range space straight R to the power of negative 1 end exponent space and space Range space straight R space equals space Dom space straight R to the power of negative 1 end exponent
For space example comma
Consider comma space straight f left parenthesis straight x right parenthesis equals straight y
straight y equals straight x squared
rightwards double arrow straight x equals square root of straight y
rightwards double arrow straight y equals square root of straight x space this space is space inverse space function space of space straight f left parenthesis straight x right parenthesis.
Hence comma space function space is space invertible. space
straight y equals square root of straight x space is space invertible space function space of space straight f left parenthesis straight x right parenthesis equals straight x squared
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Answered by Sneha shidid | 14th Jun, 2017, 09:05: AM