This is a homework question:
"Let $R$ be an equivalence relation on a set $S$. For $A ⊆ S$, we define $RA$ to be the restriction of $R$ to elements of the set $A$, i.e., $RA$ is a relation on $A$ such that for any $a, b ∈ A$, the statement $a(RA)b$ is true if and only if $a R b$ is true. Alternatively, one can define $RA$ set wise as equal to $R ∩ (A × A)$.
-Prove that for any equivalence relation $R$ on a set $S$ and subset $A$ of $S$, the restriction $RA$ is an equivalence relation on $A$.
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I have struggled to understand the above preamble. Can someone explain in lay terms what this is about?
The restriction of a function, or a relation, is the appropriate shrinking of that domain of the relation.
For example, let's say you go to a fruit shop. There are different fruits over there, say apples, oranges, bananas etc. and you define a relation on two fruits, say $a$ and $b$, call the relation $R$, given by $a R b$ if and only if $a$ and $b$ are sold by the same company. So it's possible that some apple may be related to some watermelon because they are manufactured by the same company, say Jones' company, and two bananas may not be related because one is manufactured by Reshevsky's company and the other by Wielandt's company.
Now, to restrict this relation is to apply it on a smaller subset of fruits. Let's say you don't care about whether lemons are related to apples, or bananas are related to sapota. You can "restrict" your relation to only apples, call that restriction $R_{\text{apple}}$. So you say that two apples $c$ and $d$ are related if and only if they are manufactured by the same company. However, apples are fruits of that shop, aren't they? In which case, they can be related under the same relation $R$ that was defined for all fruits in the shop. That is to say, $c R_{\text{apple}} d$ if and only if $c R d$.
So what we have done in a nutshell : The relation has not changed, but we have restricted our domain from the set of all fruits in the shop to only the set of apples. How this helps may not be immediate to see in mathematics, but here, it helps because we are focusing our attention only on apples using the new relation, and not caring about other fruits.
Please ask if you do not understand anything in this explanation or need me to elaborate.