Equivalence Relations In Real Life

The equivalence relation (ER) has the following three properties

• x ~ x, which means x is related x. This property is called reflexive.
• x ~ y, then y ~ x, This property is called symmetry.
• x ~ y, y ~ z, then x ~ z. This property is called transitive.

Not every relation is ER. Take for example knowing. Since I know myself. So knowing is reflexive. But it is neither symmetric nor transitive. Suppose I know you, it does not necessarily mean you know. And let I know you and you know somebody else. It does not necessarily mean I know that person.

However, family members is an ER. As I’m a family member of myself. If you are my family member, I’m your family member. If you are my family member and there is another person who is your family member, then that person is also my family member.