Relations And Functions

We have the function is defined for , Also, Domain of is .

We have,
... (i)
Putting , we get





Putting , in E (i), we get



Putting , in E (i), we get



Putting , in E (i), we get

is real If

The structure (N,) satisfies the closure property, associativity and commutativity but the identity element 0 does not belong to N.
So, N is a semi-group

Given Squaring both side, we have

Two events and are independent if and only if the occurrence of one event does not affect the probability of the other. This can be mathematically represented as: In terms of conditional probabilities, we have the condition for independence of events as: Thus, the correct answer is .