Types of Relations in math

There are many types of relations in maths. Understanding relations in math will help you appreciate many other math topics as well as programming. Let’s talk about 4 main types of relations. These are many-to-many relation, one-to-many relation, many-to-one relation, and one-to-one relation. We will address them one by one with examples.

Types of relations: One-to-one relation

As the name suggests, this mathematical relation is just one to one. Imagine we are sharing apples to school children and each child receives one and only one apple. In such a case we say the mathematical relation between the school children and the shared apple is one-to-one. Every school child receives one apple. Similarly, in a room full of married people. If every married person has only one spouse then we say the relation is one-to-one. One husband to one wife and one wife to one husband. One-to-one gives no room for polygamy, where a man can marry more than one wife and vice-versa.

In mathematical terms, we say that each element in the domain has only one image in the co-domain and each element in the co-domain is associated with only one element in the domain. In particular, each element of the domain has a unique image.

One-to-many Relation

Now, as the name suggests, this mathematical relation is one element to many others. What is meant by that statement? Let’s revisit our school children example. If it found out that at least one school child received more than one apple, then the relation is no longer a one-to-one relation. From then onwards it becomes a one-to-many relation. So in a one-to-many relation polygamy is allowed. This is because one person can have more that one wife or husband.

In mathematical terms, we say that one element in the domain has many images in the co-domain. That is to say, it is not all elements of the domain that has unique images.

Many-to-one relation

Okay so now let’s talk a bit about many-to-one relation. Well, I’m sure by now you can guess what this type of mathematical relation really entails. It is just as the name says. That is many elements to one. To illustrate let’s go back to our school children and the apple scenario. Now assuming 2 school children are asked to share one apple. So many school children (2 in this case) are sharing just one apple. The relation is many-to-one.

The moment the domain and co-domain of a one-to-many relation switches sides, we get a many-to-many relation.

So in a polygamous marriage, that is a marriage with more than one spouse, if the man is the one who has many wives then from the man’s point of view (ie. taking the man as an image in the domain) he is in a one-to-many relation. But from the wives’ point of view, they are in a many-to-one relation. Since they are sharing just one man.

Therefore, in mathematical terms, a many-to-many relation is a relation in which several or many elements in the domain have one image in the co-domain.

Many-to-many relation

Now finally let’s talk about many-to-many relations. As the ongoing discussions have made it clear. The name clearly defines this type of mathematical relation as well. I’m sure you got it right. This type of relation is just many elements to many other elements. Like we have done all along, let us use the school children and apple illustration to make it even clearer for us.

Furthermore, assuming in the sharing of the apples to the school children, some children (that is two or more) are allowed to share one apple and also others (that is some individuals) are allowed to have more than one apple. Now in such a scenario, we have a many-to-many relationship. Because one child can have more than one apple and one apple can be shared by more than one child.

Now with our marriage scenario, assuming the husband has many wives and at the same time, each wife can also have many husbands. Well, that will be something right? Yes, that will be a many-to-many relation. This doesn’t happen in real life or even if it does, not that I know of.

Since this one is a bit tricky, let’s use one more example to illustrate. If you enjoy movies then you will like this illustration. Now we all know that a movie actor can act in many movies. But at the same time, one movie does have many actors. So the relation between movie actors and movies is many-to-many. This is because you can find one actor in several movies and also find many actors in one movie.

So in mathematical terms, we say several elements in the domain have many elements in the co-domain. And several elements in the co-domain are associated with many elements in the domain.

Conclusion

We all have relations. They make life sweeter. As Emily Esfahani Smith puts it The Secret to a meaningful life is meaningful relationships. Similarly, relations in math make math even sweeter. Just as in real life we all admire certain relations more than others. Share in the comments which type of mathematical relation you adore most and why.

Don’t forget to do share and do justice to math.

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