21.5. Question 4

Today, we’re going to dive into the fascinating world of relations in set theory. Can anyone tell me what a symmetric relation is?

I think it’s when if (a, b) is in the relation, then (b, a) must also be in it?

Exactly, great job! We can remember this by thinking of 'if a loves b, then b loves a.' Now, what about an anti-symmetric relation?

Is it true that both (a, b) and (b, a) can only be in the relation if a equals b?

Correct! It’s like saying if there's a relationship between two different elements, then one cannot have its reciprocal in the relation. Our acronym here can be A = anti-symmetric.

Now that we understand symmetric and anti-symmetric relations, let’s look at reflexive relations. Who can explain what it means?

A relation is reflexive if all elements relate to themselves, right?

Right! For example, (a, a) for all a in set S must exist. Conversely, what’s an irreflexive relation?

None of the pairs (a, a) can be in the relation. So if (1, 1) is in R, it can’t be irreflexive?

Exactly! Irreflexivity is more restrictive. Remember that when discussing relations.

Let’s discuss relations that can be both symmetric and anti-symmetric. What do you think happens in this case?

If both must hold, I guess the only possibility is including no elements except pairs of the form (a, a)?

Exactly! The only relation is where the only pairs are of the form (a, a) or none at all. And what if we added the condition of irreflexivity?

Then we couldn’t have any pairs at all, making it the empty relation!

Think about your friendships: Can you share examples where a relationship could be reflexive or irreflexive?

If I think about it, friendships are usually reflexive since I consider people my friends if they consider me one too!

But if I think about acquaintances, that could be closer to irreflexive if neither person considers the other a friend.

Great connections! Remember, interactions in real life often mirror these mathematical concepts.