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Defintion of symmetric relation.. here X is a set and a,b are elements of X
\forall a, b \in X,\ a R b \Rightarrow \; b R a.
Defintion of transitive relation.. here X is a set and a,b and c are elements of X
\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc

Now if you see given relation S=ϕ
There are no elements in the set, so it does not violate the definitions of the symmetric and transitive realtions above..
Hence it is symmetric and transitive.

But the given relation is not reflexive...
in a reflexive relation, for every element a of set X, aRa.
but here we donot have 1R1 and 2R2 and 3R3.
So, the given relation S=ϕ is not reflexive

by Boss (31.5k points)
edited by

in fact the smallest symmetric and transitive relation is ϕ.

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