00:01
In the given question, we have to check the given relation r for different properties.
00:06
The given relation is defined as a, r, b is defined as if and only if a is related to b if and only if a plus b is even.
00:20
Now in the first part, we have to check it for the reflexivity.
00:25
Now, our relation r is said to be reflexive.
00:27
If a is related to a for all a belonging to the set a now the given relation is defined as if a plus b is even then we can say that a is related to b now we will calculate that a plus a is equal to 2a and 2a will always be a even number hence this implies a is always related to a and this will always be a even number so it is true for all all a belonging to the set.
01:02
Hence r is reflexive.
01:05
Now in set, in part b, we have to check whether r is irreflexive or not.
01:10
Since we already proved that r is reflexive, so it cannot be irreflexive.
01:15
Because irreflexive relations are those in which for all a belonging to the set we have a does not relate to a.
01:24
It means no a must be related to itself in the set.
01:30
So, r is not irreflexive.
01:34
In the third part, we have to check whether the relation is symmetric.
01:40
Now our relation is said to be symmetric if a related to b, then this implies b is also related to a.
01:49
Now let assume that a is related to b, this implies that a plus b is an even number.
02:00
Now we can say if a plus b is an even number then b plus a is also even number.
02:08
And according to the definition, this shows that b is related to a.
02:12
And this is true for all a comma b belonging to set a since we took the arbitrary ab.
02:19
Hence r is a symmetric relation.
02:23
In next part, we have to check for asymmetricity.
02:28
Our relation is set to be asymmetric if a related to a symmetric.
02:31
To b implies b is not related to a...