00:01
Hi here for the given question.
00:03
We need to find a bijection from half open interval, which is closer than 0 open it 1 to closed interval, which is 0 comma 1 now here we can define a function that maps on half open interval to closed interval.
00:36
It is a one to and a onto then it is said to be a bijection.
00:43
So here let us define a function g from 0 1 half open interval to closed interval such that for any x which belongs to 0 comma 1 the value of g of x is equal to x if x is not equal to 1 by 2 and if x is equal to 1 by 2, then g of 1 by 2 is equal to 1 now here we need to prove that this is a bijective in order to prove that first of all, we will prove that it is one to one function.
01:19
So here in order to prove that it is a one to one function here we will take for some x 1 and x 2 which belongs to half open closed interval such that g of x 1 is equal to g of x 2.
01:39
So here in our case now, we need to show that x 1 is equal to x 2 now here if both x 1 and x 2 are not equal to 1 by 2, then here we can say that g of x 1 is equal to x 1 and g of x 2 is equal to x 2.
02:05
So here we can say that the function value will be equal to as x 1 equals to x 2.
02:11
So here in our case the function will be one to one now further here if either of the one is equal to either of x 1 or x 2 is equal to 1 by 2 then here the value will not be one one.
02:41
So here the function will not be one to one one because we will not get the value of x 1 equal to x 2.
02:50
So here the value of x 1 will be not equal to x 2...