00:01
We are going to determine which of these relations are reflexive.
00:07
Variables x and y represent integers and positive means greater than 0.
00:15
In part a, the relation is defined x is related to y if and only if x times y is greater than or equal to 0.
00:25
In part b, x is related to y if and if x plus y is positive.
00:29
In part c x related to y if and only if x times y is positive part d x related to y if and only if x minus y is positive and in part e x related to y if if if if and only if x plus y 7 so let's start with part a so we want to see if it's reflexive that is if if an integer number is related to itself that will translate to x related to x if and only if x times x greater than or equal to 0 that is x related to x if and only if x squared is greater than or equal to 0 and this inequality is true for any real number in fact any real number square is positive or 0 so this is true so the relation in part a is reflexive because for any, integer x, x squared, is greater than or equal to zero.
02:07
Now let's see part b.
02:10
Now we translate what means x relate to x if and only if x plus x is positive.
02:22
That is, x plus x greater than 0.
02:27
That is, x will be related to x if and only if 2x is greater than 0.
02:40
That is x is related to x if and only if x x greater than 0 so because x can be any integer number it can be negative this is not true in general so the relation is not always true for x and into the number the relation is not reflexive for example take a negative number x equal negative 1 is not related to itself since the sum negative 1 plus negative 1 is negative 2 which is not greater than 0.
04:09
That is because the relation given in part b translate when y and x are equal translate to x plus x positive.
04:22
The sum of the integer plus itself is positive.
04:28
In this case, when x is a negative number, the sum of the number itself is two times the number which will be negative, now positive.
04:41
So in fact, we can say in this relation, any negative number is not related to itself.
04:45
It is sufficient to have one that is not related to itself for the relation not to be a reflection.
04:54
So it's not reflexive.
05:01
Part c, we have, we translate x related to x if i don't live, x times x is positive.
05:16
X times x is positive.
05:18
That's the same as x related to x...