Finding the supremum and/or infimum of these sets.. (real analysis)

I've been working on real analysis for a short time and am really struggling to find the supremum and infimum of these sets (or if they exist at all)

the set of rational numbers in the interval [0,1]

the set of real numbers in the half open interval ]0,1]

the half line x ∈ R | x≤0

S={(x^2+y^2/xy)|x>0, y>0}

thank you! :)

Re: Finding the supremum and/or infimum of these sets.. (real analysis)

Quote:

Originally Posted by

**kaya2345** I've been working on real analysis for a short time and am really struggling to find the supremum and infimum of these sets (or if they exist at all)

the set of rational numbers in the interval [0,1]

the set of real numbers in the half open interval ]0,1]

the half line x ∈ R | x≤0

S={(x^2+y^2/xy)|x>0, y>0}

The first two have both inf & sup. And they are the same for both set.

The third has only sup.

For the fourth, is that the same as $\displaystyle \frac{x}{y}+\frac{y}{x}~?$

Re: Finding the supremum and/or infimum of these sets.. (real analysis)

Quote:

Originally Posted by

**Plato** The first two have both inf & sup. And they are the same for both set.

The third has only sup.

For the fourth, is that the same as $\displaystyle \frac{x}{y}+\frac{y}{x}~?$

thank you for that :) do the first two have both because they're rational and real? would it not be different for the half open interval?

and yes the fourth is the same as x/y + y/x

Re: Finding the supremum and/or infimum of these sets.. (real analysis)

Quote:

Originally Posted by

**kaya2345** thank you for that :) do the first two have both because they're rational and real? would it not be different for the half open interval?

and yes the fourth is the same as x/y + y/x

NO. $\displaystyle \inf(0,1]=0~\&~\inf[0,1]=0$.