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An equivalence relation is a relation which “looks like” ordinary equality of numbers, but which may hold between other kinds of objects. Here are three familiar properties of equality of real numbers:
1. Every number is equal to itself: x = x for all x ∈ R.
2. Equalities can be “reversed”: If x, y ∈ R and x = y, then y = x.
3. You can “chain” equalities together: If x, y, z ∈ R and x = y and y = z, then x = z. These three properties are captured in the axioms for an equivalence relation.
Definition. An equivalence relation on a set X is a relation ∼ on X such that:
1. x ∼ x for all x ∈ X. (The relation is reflexive.)
2. If x ∼ y, then y ∼ x. (The relation is symmetric.)
3. If x ∼ y and y ∼ z, then x ∼ z. (The relation is.)
so let's switch y with x + |x|: I2x|
so the relation is reflexive.
(2) so lets check for symmetric |y|+|x| : Iy+xl
so the relation is also symmetric
(3) lets check for transitive
I x | + I y I : I x+y l.
I y | + I z I : I y+z l.
I x | + I z I : I x+z l.
so the relation is also transitive