📂Set Theory

Equivalence Relations in Mathematics

Definition ¹

A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation.

Explanation

To put the concept of an equivalence relation in non-mathematical terms, it’s like saying “it’s all the same.”

While it’s not always necessary to have a reason when studying mathematics, if there were to be a practical reason for studying mathematics, it could be said to “simplify complex concepts into easier, more manageable areas to comfortably solve problems.” In order to discuss this, one must be able to express that things are “essentially the same,” which is exactly what an equivalence relation is.

Examples of equivalence relations include $=$, $\equiv$, $\iff$, but just these examples might not fully illustrate why equivalence relations are important in mathematics. It feels too natural now to adopt a new perspective.

For instance, let’s say we have a set like $Q$. If we represent the ordered pair $(n,m)$ as $\displaystyle (n,m) = {{n} \over {m}}$, then $Q$ is essentially the same as the well-known set of rational numbers, $\mathbb{Q}$. However, upon closer inspection, we cannot say that $Q$ and $\mathbb{Q}$ are the same set because $(2,3)$ and $(4,6)$ are different elements in $Q$, whereas $\displaystyle {{2} \over {3}}$ and $\displaystyle {{4} \over {6}}$ are considered the same element in $\mathbb{Q}$. The difference between $Q$ and $\mathbb{Q}$ lies in whether there is simplification or not.

In $\mathbb{Q}$, we can discover a familiar equivalence relation. If we call this equivalence relation $\sim$, it can be defined as follows: $$ {{ a } \over { b }} \sim {{ c } \over { d }} \iff ad = bc $$ In fact, considering $\displaystyle {{2} \over {3}}$ and $\displaystyle {{4} \over {6}}$, since $2 \cdot 6 = 12 = 3 \cdot 4$, it follows that $\displaystyle {{2} \over {3}} \sim {{4} \over {6}}$, meaning the two elements can be said to be equivalent in $\mathbb{Q}$. On the other hand, without such an equivalence relation being given in $Q$, although we know $(2,3)$ and $(4,6)$ are essentially the same, we cannot expressly say so.

Thus, the process of stating that things that are originally the same as being the same may seem needless and overly theoretical at first glance. Our intuition already knows they are the same, but it feels like we’re saying the same thing again using difficult language and symbols. However, rigor is considered a supreme value in mathematics, and this process is unavoidable while simultaneously, it can be used to break through the limits of mathematics. For instance, consider bending a line segment and joining the ends together as shown in the picture. When it was a line segment, the two points were clearly different. However, by applying an equivalence relation and treating the two points as essentially the same, a completely new loop is created. The field of topology studies such phenomena, expressing the act of “joining” in mathematical terms through the use of equivalence relations.