Answer to Question #280833 in Discrete Mathematics for ainina

Question

Define a relation - on the set of real numbers by

x-ymeans I x | + I y I : I x+y l.

which of the properties for an equivalence relation does - satisfy?

Answer 1

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