Let us look at the concept of a "race" in terms of the concept of an equivalence relation.

What is a "race" as in part of the "human race"?

At one time, a "race" meant a nationality is in the following.

• The Irish race
• The Jewish race
• The German race
• ... and so on ...

In recent times, a "race" has come to mean a differentiation based on the color of one's skin such as the following.

• The Caucasian race
• The African-American race
• The Indian race (from India?)
• The Native-American race
• The Hispanic race
• ... and so on ...

From what we know about equivalence relations, does the relation "is the same race as" meet the conditions of an equivalence relation? If so, then we should be able to use that relation to determine racial groups.

To do so, we need to verify three properties: reflexivity, symmetry, and transitivity.

Note: Should you prefer to not discuss race in terms of equivalence relations, another interesting approach is the relation "speaks the same language as".

Is the relation R as "is the same race as" reflexive?

Is someone the same race as them-self? Yes.

Is the relation R as "is the same race as" symmetric?

If you are the same race as someone else, is that someone else the same race as you? Yes. If not, explain why.

Is the relation R as "is the same race as" transitive?

If you are the same race as someone else, and they are the same race as another person, then are you the same race as that other person. Yes. If not, explain why.

If "is the same race as" is an equivalence relation, then we can divide all humans into racial groups. Thus, if there is any discrimination between groups, we can address that. We can define hate crimes between racial groups.

Note that the fuzzy and imprecise and shades of gray between racial groups should disappear when we use the concept of an equivalence relation.

How can me define the relation "is the same race as"?

If we cannot make a precise definition, then one or both of the following implications result.

In the one extreme, if we need to continually make the definition more specific, then we reach the following limit.

• Everyone is of a distinct race, different in race than anyone else, and the number of races is equal to the number of human beings. Thus, every crime could be considered a hate crime and any inequity in treatment between individuals is discrimination in some form.

At the other extreme, if we need to continually make the definition less specific and more general, then we reach the following limit.

• Everyone is of the same race. No crime between individuals, though sad, is a hate crime, just a crime. Every interaction between individuals is between individuals of the same race.

This follows from the following equivalence relation properties from mathematics.

• Reflexive You are the same race as yourself.
• Symmetric: If you are the same race as someone else, that person is the same race as you.
• Transitive: If you are the same race as someone else, and that person is the same race as another person (three different people) then you are the same races as that third person.

Given these relations, everyone can be precisely and unambiguously grouped into distinct racial groups. Such relation processing is part of fundamental algorithms in computer science

How can we logically define racial laws, crimes, etc.?

If we cannot do this, then one observation or conclusion is that any racial laws, crimes, etc., are just not mathematically logical. If so, then using logic and rational reason on those who are using illogical and irrational reasoning may not be very productive.