machine precision and numerical cancellation

i have a question which is : "let x and y be two machine numbers working with a certain arithmetic with machine precision (mp) then" :

1) x+em=x for all x

2) x> em for all x

3) for all x and y the operation x-y is not affected by numerical cancellation

4) x-y is a machine number for all x and y

I'm torn between the first and the second answer so can someone tell me which one and why please?

thanks

[Steven Lord]

What is em? Is it related to mp at all?

Since this sounds like a homework or exam question, why don't you explain the reason you think each answer is correct or incorrect and we may be able to offer some feedback.

[sadek misto kirdi]

ah sorry mp is the same as em.

as i learned that matlab stores numbers with 16 digits of precision so in subtraction we might end up losing some digits which lead to numerical cancellation so 3) is wrong and case 4 is similar to case 3 so i was left with 1) and 2).

as i know also mp= epsilon/2 so i tried to add any machine number to eps/2 on matlab and i was getting the same number as the machine number so thats why i think 1) is true but i am not sure.

[David Goodmanson]

Hi sadek,

could you explain your reasoning for why case 4 is 'similar' to case 3?

[sadek misto kirdi]

because matlab cant recognize a difference less than eps

for example take x=1 and y=x+ eps/2

y-x will give 0 although it is not 0

[David Goodmanson]

well, is case 4 asking about the accuracy of x-y?

[sadek misto kirdi]

sorry i didnt understand your question