Proving X + Y / 2 is Between X & Y in R

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Homework Help Overview

The discussion revolves around proving that for real numbers X and Y, where X is less than Y, there exists a real number Z such that X < Z < Y. The original poster attempts to show that the expression (X + Y) / 2 satisfies this condition.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster is trying to demonstrate that (X + Y) / 2 is greater than X and less than Y, but is encountering difficulties in connecting these inequalities. Another participant suggests adding a value to both sides of the inequality for clarity. There is also a note about the importance of proper bracketing in the expression.

Discussion Status

The conversation is ongoing, with some participants questioning the validity of the original poster's approach and suggesting alternative methods. There is no explicit consensus on the correctness of the reasoning presented, and multiple interpretations of the expressions are being explored.

Contextual Notes

Participants are discussing the implications of the order of operations in mathematical expressions and the necessity of clear notation. There is an emphasis on ensuring that the expressions used correctly represent the relationships between X, Y, and Z.

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Problem:

"Suppose X, Y in R (Real Numbers), X < Y prove there exists Z in R such that X < Z < R."

I'm currently trying to prove that X + Y / 2 satisfies this but I'm getting stuck. I first show that X + Y / 2 cannot be = to either X or Y. I then try to show that X + Y / 2 is > X since X < Y but I cannot seem to tie this in. Any help would be appreciated.
 
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Nvm I think I got it.

Since X < Y
Y - X must be in Positive Reals
2 is in Positive Reals thus 1/2 is in Positive Reals
Y - X / 2 is thus in Positive Reals
You can then rearrange such that you can the equation

X + Y / 2 - X in positive reals
Thus X < X + Y / 2.

Right?
 
Dunno about that (you ought to bracket things up), but if x<y why don't you just add something to both sides?
 
matt grime said:
(you ought to bracket things up)
In case you missed it, matt's point is: x+ y/2 is NOT between x and y:
(x+ y)/2 IS!
 

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