PLEASE SHOW ALL OF THE WORK Prove or disprove For all positi

PLEASE SHOW ALL OF THE WORK!!!!!!

Prove or disprove: For all positive real numbers x and y,

Solution

this statement is false .Lets understand this with an example.

here [] represents the greatest integer function

that is [a] = greatest integer function a , if the domain of a E [c,d) then the range of a will be = c . Here c and d are real numbers and c in particular is an integer value.

lets start with the example:

let x = 4.5 and y = 3.1

=> [x] = [4.5] = 4

[y] = [3.1] = 3

=> [x] . [y] = 4 * 3 = 12

[x*y] = [4.5*3.1] = [13.95] = 13

so for this particular example we can see that [x.y] > [x].[y]

but lets take another example if x = .5 and y = .5

the [x] = [.5] = 0 and [y] = [.5] = 0

=> [x].[y] = 0

but [x.y] = [.5 . .5] = [.25] = 0

=> [x].[y] = [x.y]

now lets take another example

let x = 3 and y = 2

=> [3].[2] = 3.2 = 6

and [3.2] = [6] = 6

so for all positive integers

[x.y] = [x].[y]

but for the rest of the positive real numbers

[x . y] <= [x] . [y] will not always be true as we hve seen with the above examples.

Hence the statement is false.