Problems from Introduction to intervals

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.

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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.


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.

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