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Can someone help me in "part b" of this question- https://gateoverflow.in/1724/gate1998-10 .
I am still not able to understand why $R^0$ is considered here ? 
and what is $R^0 $?
Is it Equality relation?
Do we have to consider it in every question of this type ?

 

in Set Theory & Algebra by Boss (16k points) | 185 views

1 Answer

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In relations $R^0$ is Always the Identity relation (Equality relation as you termed). We can prove it in 3 ways.

1. Just By-heart it and think of it as an Axiom of relations.

2. We know in Relation theory $R^n.R^0 = R^n$ where $.$ is Composition Operator (Composition/Composite of two relations)

We could even write $R.R^0 = R$, Now you need to think what should be $R^0$ in such a way that When any relation is Composed with it, results in the same relation. (Sounds like Identity element for Composition of relation)

3. Answer Why in General mathematics $n^0 = 1$ where $n \neq 0$. If you can answer that, You can answer why $R^0$ is always the Identity relation. 

Try it.

by Boss (26.3k points)
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+1
Thanks for point $2$ and $3$. Now it looks intuitive to me. :)

$\frac{R^1}{R^1} = R^{1-1}= R^0= R^{-1}.R = I$

Matrix for $R^0$ will be = $\begin{bmatrix} 1 &0 &0 \\ 0&1 &0 \\ 0&0 &1 \end{bmatrix}$ for n=3 and it's always true for every non empty relation.
0
Nice. Preparing for GATE 2019??
0
Yes.. Congratulations for AIR 67. :)
0
Thank you. And All the very best :) keep doing nice work.
0
I'll try my best. Thank you :)
0
haha @Soumya u take my words :p

Congrats AIR 67:) Going IITB or IISC?
0
haha @srestha  :P (y)
0
Fingers crossed for IISc. Currently in IITB (Fees submitted) :P ..So, Partially your college-mate. =D
0
hey I am not that Shrestha. I am an aspirant.

All d best. Keep rocking:)
0
Okay. My bad. A misunderstanding. I apologize.
+1
Are aisa hote hai. I never mind :)
0

I didn't understand this part " LCM of 3 and 5 is 15. So, m=0 and n=15"

can you please explain, how to get m and n

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