Let R and S be two equivalence relations on a set A. Then
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Given, R and S are relations on set A.
∴ ∴ 
and 
⇒ ⇒ 
⇒ ⇒ 
is also a relation on A.
Reflexivity: Let a be an arbitrary element of A. Then, 
⇒ ⇒ 
and 
,
[ 
R and S are reflexive]
⇒ ⇒ 
Thus, 
for all 
.
So, 
is a reflexive relation on A.
Symmetry: Let 
such that 
.
Then, 
⇒ ⇒ 
and 
⇒ ⇒ 
and 
,
[ 
R and S are symmetric]
⇒ ⇒ 
Thus, 
⇒ ⇒ 
for all 
.
So, 
is symmetric on A.
Transitivity : Let 
such that 
and 
. Then, 
and 
⇒ ⇒ 
and 
⇒ ⇒ 
and 
⇒ ⇒ 
and 
⇒ ⇒ 
Thus, 
and 
.
So, 
is transitive on A.
Hence, R is an equivalence relation on A.
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