We have to prove intersection of
equivalence relations on a set is equivalence
Let us suppose R and S are 2 equivalence
relations on a set A.
So set A implies R∩S is also equivalence
xϵA⇒(x,xϵR) and (x,xϵS)
(x,x)ϵR∩S
∴R∩S is Reflexive
Now (x,y)ϵR∩S
(x,y)ϵR and (x,y)ϵS
(x,y)ϵR are equivalence they are symmetric also.
(y,x)ϵR and (y,x)ϵS.
(y,x)ϵR∩S
∴R∩S is symmetric.
finally (x,y)ϵR∩S and (y,z)ϵR∩S
(x,y)ϵR and (y,z)ϵR⇒(x,z)ϵR
(x,y)ϵS and (y,z)ϵS⇒(x,z)ϵS
∴R is transitive and 's' is transitive
∴R∩S is transitive
∴R∩S is equivalence relation.
i.e, Intersection of equivalence relation set
is equivalence.