Question

Let the set T be given by T = x^2 < 2. Describe maximum, minimum, supremum (least upper bound), and infimum (greatest lower bound) for T (they may not exist).

Answer 1

• The supremum is
  `√(2)`.
• The infimum is
  `-√(2)`.
• There is no maximum.
• There is no minimum.

Explanation:

We have the set
`T=\{x\in\mathbb{R}:x^2<2\}`. Now, let us recall that
`√(x^2)=|x|`, and the inequality
`x^2<2` is equivalent to
`|x|<√(2)`, so our set can be written as

`T=\{x\in\mathbb{R}:|x|<√(2)\}`.

The inequality
`|x|<√(2)` is equivalent to
`-√(2)<x<√(2)`. So,

`T=\{x\in\mathbb{R}:-√(2)<x<√(2)\}`. Thus,
`T=(-2,2)`.

Now, we have that
`T` is the open interval (-2,2). From this we can extract all the information we need:

• The supremum is
  `√(2)`.
• The infimum is
  `-√(2)`.
• There is no maximum, because the interval is open.
• There is no minimum, because the interval is open.