Question

Prove that intersection of equivalence relations on a set is also an equivalence relation.

Answer 1

Hint: An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. For the objects where
a=a (reflexive property)
if a=b and b=a(symmetric property)
if a=b and b=c, then a=c (transitive property)
In this question, to prove that the intersection of equivalence relations on a set is also an equivalence relation proves all that the set satisfies all the properties of equivalence.

Complete step-by-step solution
Let us assume P and Q are two equivalence sets whose intersection is to be proved of set S
So set S implies $$P\cap Q$$ equivalence, where
$$x\in S\Rightarrow \left(x,x\in P\right)$$and$$x\in S\Rightarrow \left(x,x\in Q\right)$$
$$\left(x,x\right)\in P\cap Q$$
$$\therefore P\cap Q$$is Reflexive
Now to check symmetric
$$\left(x,y\right)\in P\cap Q$$
Hence
$$\left(x,y\right)\in P$$and $$\left(x,y\right)\in Q$$
are symmetrical
$$\left(y,x\right)\in P$$and $$\left(y,x\right)\in Q$$
$$\left(y,x\right)\in P\cap Q$$
Hence $$P\cap Q$$is symmetric.
Now for transitive property
$$\left(x,y\right)\in P\cap Q$$and $$\left(y,z\right)\in P\cap Q$$
$$\left(x,y\right)\in P$$and$$\left(y,z\right)\in P$$$$\Rightarrow \left(x,z\right)\in P$$
$$\left(x,y\right)\in Q$$and$$\left(y,z\right)\in Q$$$$\Rightarrow \left(x,z\right)\in Q$$
Therefore P and Q are transitive
$$P\cap Q$$is transitive
Hence all the properties of equivalence are satisfied, therefore $$P\cap Q$$is an equivalence relation.

Note: Two elements from an equivalence relation are called equivalent. The set of one element has only one equivalence relation with one equivalence class.