Problems from Introduction to intervals

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

a) $A = {x | x \leq 3}$

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

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

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

e) $N$

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

See development and solution

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 $B = {x | x = 2^{k}, k \in N}$ 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 $x \leq 1 \forall x \in C$ .

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.

Hide solution and development