Let f and g be two binary relations on a set a we define the...

Let f and g be two binary relations on a set A. We define the binary relation h = f ⊗ g on A as follows: h(x, y) = min(f(x, y), g(x, y)). Show that x has the relation h to y if and only if x has the relation f and the relation g to y.

Transcript

00:01 In this question, we know that a relation is said to be reflexive if and only if a, b is belonging to r which implies that b, a is also belonging to r.

00:39 So if a relation is said to be transitive relation, that is if a, b and b, c belonging to r means which implies then a, c is belonging to r.

00:57 So here we are going to consider a relation that is a less than is equal to 2b.

01:03 So from this we can able to tell for any integer a less than is equal to 2a is strictly false for the integer a is equal to minus 1 because we have minus 1 is less than is equal to minus 2 which is false.

01:22 Therefore, i can able to tell a, a is does not belonging to r.

01:27 So this is not a reflexive.

01:29 So it is false.

01:31 Now consider a region 1, 3 that is 1 is less than is equal to 2 times of 3.

01:39 So this implies 1 less than is equal to 3.

01:42 So this is true statement.

01:44 But if i am writing in the other side, 3 is less than is equal to 2 times of 1.

01:49 So this is the wrong statement.

01:52 So your symmetric relation is not satisfied.

01:58 Now let's move on to third part.

02:00 That is a less than is equal to 2b, b less than is equal to 2c.

02:05 Then i can able to tell a is less than is equal to 2 times of 2c.