Say which of the following sets are bounded and which are unbounded:

a) $$A=\{x \ | \ x\leq3 \}$$

b) $$B=\{x \ | \ x \ \text{ is a positive power of } 2 \}$$

c) $$C=\{x \ | \ x=2 \text{ or } x=5 \}$$

d) $$D=\{x \ | \ 0 < x < 1 \}$$

e) $$\mathbb{N}$$

Besides that, write as intervals the sets that admit this notation.

### Development:

a) Note that $$A = (-\infty,3]$$ and using the definition of bounded set, a set is bounded if the absolute value of all its elements is less than or equal to a certain number. In this case, since the interval has no lower endpoint, it cannot be a bounded set.

b) The set is $$B =\{x \ | \ x=2^k, \ k\in\mathbb{N}\}$$ and since $$k$$ can be any natural number, the set $$B$$ is unbounded.

c) $$C = \{2,5\}$$ and, therefore, taking $$M = 5$$, we see that $$C$$ is a bounded set since $$2,5 \leq 5$$.

d) Set $$D$$ can be rewwritten as $$D = (0,1)$$. It is also a closed set because if $$M = 1$$, it is satisfied that $$x\leq1 \ \forall x\in C$$.

e) The set of the natural numbers is unbounded because there is no such positive number that all naturals are less than or equal to.

### Solution:

a) Unbounded, and can be written as $$A = (-\infty,3]$$.

b) Unbounded.

c) Bounded.

d) Bounded and it is possible to write it like $$D = (0,1)$$.

e) Unbounded.