00:01
Okay, we want to figure out a bunch of stuff about these relations.
00:04
Let's just remind ourselves of a few definitions.
00:06
So reflexive means something is related to itself.
00:12
Symmetric means if two things are related in one order, they're related in the other order.
00:27
And then transitive means that if we have a, relation b and b relation c, then we have a relation.
00:40
Is related to c.
00:42
And then irreflexive is the opposite of reflexive.
00:47
So x is never related to itself.
00:50
And then antisymmetric is the opposite of symmetric.
00:55
So that means whenever x is related to y, y is not related to x.
01:04
Okay.
01:05
So let's figure out what's true about these.
01:10
So first relation, part a is that x and y have a common prime factor.
01:17
X and y have a common prime factor.
01:22
Well, is it reflexive? one and one do not have a common prime factor.
01:36
I need to find two things that are the same that are not.
01:40
So like, it's not reflexive.
01:44
But it's not irreflexive either because like two and two do have a common prime factor.
01:50
So that's good, that's bad.
01:52
So it's neither reflexive nor irreflexive.
01:56
It's not never true that something is related to itself, but it's also not always true.
02:02
It's most of them true.
02:04
Okay.
02:05
What about symmetric? yeah, it's symmetric.
02:08
If two things have a common prime factor, they have a common prime factor.
02:11
I don't care what order i list this is.
02:13
So it is symmetric, but it's not anti -symmetric.
02:19
We certainly can't be both.
02:22
And then is it transitive? if x and y have a common prime factor and y and z have a common prime factor, do x and z have a common prime factor? i don't think so, because i can say like four and six have a common prime factor and six and nine have a common prime factor, but four and nine don't.
02:43
So this one's not.
02:45
Transitive.
02:49
So the only thing we check is symmetric in that case.
02:52
Okay, what about part b? x minus y is a multiple of three.
03:01
I'm going to write equals 3k here, where k is an integer.
03:06
Okay, let's see.
03:10
Reflexive.
03:11
So is a number minus itself always a multiple of three? a number minus itself is always zero.
03:16
So yes, it's definitely reflexive.
03:18
Zero is a multiple of three, which means it can't be irreflexive.
03:25
What about symmetric? if x minus y is a multiple of three is y minus x a multiple of three, yes, negative of a multiple three is a multiple of three.
03:36
So it is symmetric, which means it's not antisymmetric...