Definition of an equivalence relation:

An equivalence relation on a set is a relation that satisfies three properties:

1. Reflexivity: Every element in the set is related to itself. For example, if “a” is an element in the set, then “a” is related to “a”.

2. Symmetry: If one element is related to another, then the second element is also related to the first. If “a” is related to “b”, then “b” is also related to “a”.

3. Transitivity: If an element is related to a second element, and the second element is related to a third, then the first element is related to the third. If “a” is related to “b”, and “b” is related to “c”, then “a” is related to “c”.

If a relation satisfies all three of these properties, it is called an equivalence relation.

Example of an equivalence relation on the set of real numbers:

Consider the relation on the set of real numbers “R”, defined as follows:

“a is related to b” if and only if “a – b” is an integer, meaning the difference between “a” and “b” is an integer.

Let’s check the three properties:

1. Reflexivity: For any real number “a”, “a – a = 0”, which is an integer. So, “a is related to a” for all real numbers.

2. Symmetry: If “a is related to b”, then “a – b” is an integer, and thus “b – a = -(a – b)” is also an integer. So, if “a is related to b”, then “b is related to a”.

3. Transitivity: If “a is related to b” and “b is related to c”, then “a – b” and “b – c” are both integers. Adding them gives “a – c”, which is also an integer. So, if “a is related to b” and “b is related to c”, then “a is related to c”.

Since the relation satisfies all three properties, it is an equivalence relation.

Interpretation:
This relation groups real numbers into equivalence classes based on their fractional parts. For example, “1 is related to 4” because “1 – 4 = -3”, which is an integer, and both numbers have the same fractional part. Similarly, “2.5 is related to 5.5” because “2.5 – 5.5 = -3”, which is an integer.