Find the supremum and infimum of each S.

S  = {x | x  <10}  
S  = {x | x  > -5}

Solution:

To find the supremum and infimum of each set S, let's consider the given sets:

For the set S = {x | x < 10}:

The supremum (or the least upper bound) is 10, as there is no element in the set that is greater than 10, but any value less than 10 can be an upper bound.

The infimum (or the greatest lower bound) does not exist in this set. There is no element in the set that is a lower bound, but any value less than 10 can serve as a lower bound.

For the set S = {x | x > -5}:

The supremum does not exist in this set. There is no element in the set that is an upper bound, but any value greater than -5 can be an upper bound.

The infimum (or the greatest lower bound) is -5, as there is no element in the set that is less than -5, but any value greater than -5 can serve as a lower bound.

In summary:

For S = {x | x < 10}: Supremum = 10, Infimum does not exist.

For S = {x | x > -5}: Supremum does not exist, Infimum = -5.