00:01
Consider the following relations on the set of positive integers.
00:07
R1 is the set of pairs x, y, both x and y positive integers such that x plus y is greater than 10.
00:19
R2 is the order pairs x, y such that y, the second element, divides x, the first element.
00:29
R3 is the pairs x, y such that the greatest common divisor of x and y is 1.
00:37
And r4 is the pairs x, y such that x and y have the same prime divisors.
00:48
So we want to know which of these relations are reflexive.
00:54
Remember reflexive relation is a relation that or in which any element is related to itself.
01:08
So in this case, for example, let's see r1.
01:14
In r1, a pair is in the relation if x plus y is greater than 10.
01:21
So we want to see if any positive integer is related to itself.
01:28
That is, if the sum of the positive integer and itself, which is 2 times the positive integer, is greater than 10.
01:37
That is true only for x greater than 5.
01:40
That is, for example, 1 is not related to itself because 1 plus 1 is not greater than 10.
01:50
So this is not a reflexive relation.
01:59
So it is not reflexive.
02:02
It is sufficient that there is a number in the set where the relation is defined such that it is the order pair with the same elements.
02:16
It is not in the relation.
02:18
That is sufficient to say that the relation is not reflexive.
02:24
So i can say this way, it is not reflexive since, for example, because there are more cases, but i'm going to put only one case.
02:34
1, 1 is not in r1.
02:41
So saying that any element in the set is related to itself means, in terms of the set of pairs, that we must find all possible pairs of equal components.
02:59
In this case, because the relation is defined in the positive integers, we get to find all order pairs with equal positive integer elements.
03:13
For example, 1, 1.
03:14
But 1, 1 is not in r1, the relation of r1, because 1 plus 1, the sum of the components, is not greater than 10.
03:28
So i can say, because, to emphasize only, because 1 plus 1 is not greater than 10.
03:49
So that the pair 1, 1 is not in the set r1 because the condition that defines the pairs that are in r1 is not satisfied by the pair 1, 1.
04:02
All the pairs that are not in r1 are 2, 2, 3, 3 and 4, 4.
04:09
Of course, from then on, from 6 on, all pairs are in r1.
04:18
But remember, it is sufficient that there is an element that is not related to itself to say that the relation is not reflexive.
04:29
So we have found this, at least one element that is not related to itself, for example 1 here, number 1.
04:40
Ok, that is r1.
04:44
Now say r2 is defined as a pair is in the relation if the second component divides the first.
04:52
We want to know if any element divides itself, and that is true.
05:00
So it is reflexive.
05:02
And the reason why is any positive integer divides itself...