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in Set Theory & Algebra by (7 points) | 231 views

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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.

Extra:
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 (31.2k points)
edited by
+1

in fact the smallest symmetric and transitive relation is ϕ.

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