Equivalence

Introduction

When we say two objects are equivalent, we mean that for all of our intents and purposes, the two objects are the same. For example, two apples are equivalent because they taste like… apples. In mathematics, there exist many forms of equivalence, and with it we can make bizarre statements such as 0 being equivalent to 1.

In order to define an equivalence relation within a set of numbers, we must satisfy 3 essential properties: reflexivity, symmetry, and transitivity. That is, if we take any three elements a,b,c in the set with some relation \sim, we have

Reflexivity: a\sim a

Symmetry: a\sim b\Rightarrow b\sim a

Transitivity: a\sim b, b\sim c\Rightarrow a\sim c

If the relation \sim satisfies all three properties, then it is considered an equivalence relation. The set of all elements that are equivalent to some element is called its equivalence class, and can be represented by any element in the set.

Exercise: Which property of equivalence implies that any element can represent its equivalence class?

The set of equivalence classes is known as the quotient set or quotient space, and has wide applications in ring theory.

Examples

In the integers, we can define a\sim b to be when a=b. Since there is only one element that equals itself for each integer, the equivalence class of each integer is just the set containing that integer.

But how do we make 0 equivalent to 1? We can try to define a relation a\sim b to be when a+1=b in the integers, but this quickly falls apart, as it doesn’t satisfy any of the three required properties.

Exercise: Check that this relation doesn’t satisfy any of the relations.

Thus we turn to an easy alternative: we just define a\sim b to be when a,b are both integers. This satisfies all three properties, so it is an equivalence relation. And clearly we have 0\sim 1 under this relation as well. There would be only one equivalence class consisting of exactly the integers themselves.

Now suppose we define a\sim b to be when a,b leave the same remainder when divided by some m. We quickly see that

    \[\dots,-2m,-m,0,m,2m,\dots\]

    \[\dots,-2m+1,-m+1,1,m+1,2m+1,\dots\]

are equivalent to each other; these in fact are the equivalence classes corresponding to this relation. How many equivalence classes are there? Since a remainder must be at least 0 and at most m, we theorize that there are exactly m equivalence classes.

Exercise: Prove that these equivalence classes are disjoint from each other.

Applications

Equivalence is a common theme in set theory, which plays into a lot of the different topics within algebra and number theory. It is the very essentials in math, and because of it we can use our favorite method of substitution in our everyday mathematical lives.

Prerequisites/What you need to know

There are no prerequisites, but a knowledge of at least Algebra 1 is recommended to fully grasp the contents of the article.

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