00:01
In this question, we are required to check the asymmetric property for those eight relationships.
00:12
First, let's record the definition of the asymmetric property for a given relationship r.
00:18
We see r is asymmetric if and only if once we have the ordered pair ab is contained in this relationship, then we must have ba is also contained in this relationship.
00:47
So to check the asymmetric property, we only need to swap the position of a and b.
00:55
Once ba also satisfies our relationship, we say this relationship r is asymmetric.
01:06
Okay, let's check them first one by one.
01:09
Okay, first, xy is in r if x plus y is equal to 0.
01:23
For the first one, let's just write down the representation of the relationship explicitly, and we will not do that for the other one.
01:34
Okay, just as what i said, now for any xy in this relationship, we know x plus y is equal to 0.
01:44
And let's check yx.
01:49
Yx means y plus x is equal to 0.
01:54
Okay, from this, we know we must have this relationship.
01:58
And this equation just tells us yx is also contained in r.
02:04
That means it's easy, right? rigorously, we need to check this relationship in this way.
02:15
But as we only have x and y in all other relationships, we only need to swap the position to check whether we still have the inequalities or inequalities for all of other situations.
02:29
Okay, the first one is easy.
02:31
Let's do the same thing for the other.
02:33
Now suppose x is equal to positive n minus y.
02:37
We know y will be equal to minus or positive x because we can multiply minus 1 on both sides.
02:46
This tells us xy, from xy in r, we know yx is also contained in r.
02:59
That means this relationship is also asymmetric.
03:03
Now let's consider the third one.
03:05
The third one tells us xy is contained in our relationship only if x minus y is a rational number.
03:17
Suppose xy is in r.
03:22
We know x minus y will be equal to r, and r is some rational number.
03:30
And what can we conclude from this equation? then we know y minus x is equal to minus r.
03:37
Because r is a rational number, minus r must be also a rational number.
03:42
That means if we have xy is contained in r, yx must also be contained in r.
03:52
This one is also asymmetric.
04:00
Okay, now let's consider the fourth one.
04:04
Xy is contained in r if and only if x is equal to 2 minus y.
04:11
Now suppose xy is contained in r.
04:13
We want to consider yx.
04:16
If yx is contained in r, then y must be equal to 2x.
04:26
But here, from x is equal to 2y, we cannot conclude that y is equal to 2x.
04:38
That means this one is not asymmetric.
04:44
For example, for 2y, we know x is equal to 2 and y is equal to 1.
04:52
It's contained in our relationship.
04:54
But if we swap the order, we'll get 1, 2.
05:02
And it's easy for us to see.
05:03
Now x is not equal to 2y.
05:05
That means this guy is not an asymmetric relationship.
05:09
Now let's consider the fifth...