00:01
Hello there.
00:03
So for this exercise we need to check some relations some operations on relations.
00:13
Okay, so let's suppose that we got some set a.
00:21
Okay, so we got some set a and we got two relations there some the relations are and the relations s and both are reflexive.
00:36
So given this we should check that the following sets corresponds that the following relations are also reflective.
00:48
Okay, so let's start.
00:50
So first we should check that if we take our union as is reflexive or not on a.
01:02
Okay, so let's recall what's the name of a union in this case.
01:07
So if there is some element, so the definition of reflexive first.
01:13
So it is say that the relation r is reflexive if on a if the relation a is on r is on r for every element a in a okay so this is basically the definition so now we are considering the union of two reflexive sets so r union s is defined as all them all all the elements that are either in r or, so it is all, it corresponds to all the elements e, such that e are on r or e, sorry, are on s.
02:32
Okay, so that's like the basic definition, but in this case this e corresponds to this tuples, a, and here a okay so this is the definition so it means that we can if an element a a is on r union s it implies that it is either on r or either on s and we know that if it is on r it is reflexive or this element is on s is also reflexive so that implies that this set r union s is also reflective, is also reflexive, sorry, missing concept, is also reflexive on a.
03:51
Then we should check the intersection, okay? so the intersection is defined as all the topos, a, such that a, a, r on r, and a, a, r on s.
04:21
Okay? so if we choose some element a on a, then we know that that a will be on r, will be a relation, okay, that is reflexive, and also will be on s.
04:51
Case because r and s are a reflexive relation on a so this is satisfied and then it is easy to see that if we choose some element a a on the intersection it will satisfy both it will be an element of r and s at the same time and both so that implies that the intersection is also reflective, reflexive.
05:25
It's also reflexive.
05:45
Let's continue.
05:48
Now we should check the symmetric difference.
05:54
So see, we should check this.
05:57
This symbol here represents the symmetric difference.
06:01
So i'm going to explain that first.
06:05
So what it means to be a reflective, a symmetric difference.
06:11
So let's suppose that you got two sets, let's say as, then a symmetric difference s corresponds to the set all the elements that are on a and all the elements that are on s, but not no in the intersection.
06:34
So that means that we are not considering these elements here.
06:38
So these elements are taking out.
06:44
Okay, so i'm going to left that picture there.
06:48
I'm going to say that now this is equal to r and s just to illustrate a little bit...