Let R be a non trivial relation on set X . If R is symmetric and anti symmetric then R is
a ) reflexive
b ) transitive
c ) equivalence
d ) diagonal relation
Actually I am confused about definition of diagonal relation i.e whether diagonal relation contains ( x ,x ) for every x belongs to Set or it contains ( x ,x ) for some x belongs to Set . And whether there is any difference between reflexive and diagonal relation ?
[Math] Diagonal relation and reflexive relation
equivalence-relationsrelations
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Best Answer
Given that $R$ is simultaneously symmetric and antisymmetric, it follows that $R\subseteq\{(x,x)~:~x\in X\}\subseteq X\times X$
To see why this is:...
A reflexive relation is one where $\{(x,x)~:~x\in X\}\subseteq R$. That is to say, it contains every possible pair of the form $(x,x)$ along with possibly containing other types of pairs too.
A diagonal relation is one where $R\subseteq\{(x,x)~:~x\in X\}$. That is to say, it contains only pairs of the form $(x,x)$, possibly not all of them, and contains no other pairs of a different form.
It is possible for a diagonal relation to be reflexive and vice versa, but only in the case that the relation is specifically $R=\{(x,x)~:~x\in X\}$
We have as a result:
Regarding reflexivity:
Regarding transitivity:
Regarding equivalence:
Regarding diagonality: