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In my textbook it says

A relation R in a set is a set of ordered pairs, so any subset of a set of ordered pairs will be a relation. This includes the empty set which is referred to as the empty relation.

What is this mean? empty set = empty relation? and if so then why empty relation is symmetric as well as transitive? My teacher said it is done by default but i still can not understand it.

In case of equivalence class i understand the concept and i thought it was easy. An equivalnce relation on Set partitions Set into subsets which are called equivalence classes. Good enough not difficult. But in the example where

A = {1, 2, 3, 4} and define a relation R by: xRy <-> x + y is even.

The relation is equivalence relation. R = {(1, 1), (1, 3), (3, 3), (3, 1), (2, 2), (4, 4), (4, 2)}

and they say equivalence classes : {1, 3} and {2, 4}

This i can not understand.. why only these two? doesnt {1 ,1} {2, 2} {3, 3} {4, 4} counted?

Set theory is too hard..I thought i understood everything and when i look back everything is new! well that must be because i couldnt have enough time for revision but this is too much.

Any definitions of empty relation and equivalence class that can be understood by person like me?