supremum and infimum

[shelly89]

Find the supremum, infimum, maximum and minimum of the following sets, or indicate that they don’t exist.


not sure about the other three, but is the first one,

min = 0
max = 2
inf = does not exits
sup = 2?

 

[Romsek]

Find the supremum, infimum, maximum and minimum of the following sets, or indicate that they don’t exist.


not sure about the other three, but is the first one,

min = 0
max = 2
inf = does not exits
sup = 2?

It doesn't look as if you've quite got it yet. Let's look at the first set {x a Natural number | x + x²> 20}

The minimum and maximum are a bit easier to understand so look at them first.

Is there a maximum x such that x + x² > 20? If x satisfies that doesn't x+1?

For the minimum just find out what's the lowest x that satisfies x + x² > 20. Is it 4?

The supremum is the least upper bound on the set. Is the set bounded? was there a maximum? Can there be a least upper bound?

The infimum is the greatest lower bound of the set. Does the set have a lower bound? Is x > 0 for all x in the set? How about 1? Can you find the greatest lower bound?

Give it another try.

 

[HallsofIvy]

0 does NOT satisfy \(\displaystyle 0+ 0^2> 22\) so is not in the set of all integers, x, such that \(\displaystyle x+ x^2> 22\)
so certainly is NOT the smallest number in that set!

It should be obvious that arbitrarily large integers will satisfy \(\displaystyle x+ x^2> 22\) so there is NO largest membe (no maximum) and no supremum.

Since this set does have a smallest member, it does have an infimum which is the smallest member.

\(\displaystyle 1+ 1^2= 2\)
\(\displaystyle 2+ 2^2= 6\)
\(\displaystyle 3+ 3^2= 12\)
\(\displaystyle 4+ 4^2= 20\)
\(\displaystyle 6+ 5^2= 25\)

Surely you could do that?