Showing if these sets are finite, countably infinite or even uncountable

Question

I am working on a few questions. I'm trying to see whether the following are finite, uncountable or countably infinite. Explanations using surjective and injective would help me understand at a deeper level.

1. $\{1/m \mid m ∈ \mathbb{Z}, m ≠ 0\}$

2. $\mathbb{R} - \mathbb{N}$ (Real numbers - Natural numbers?)

3. $\{x \mid x ∈ \mathbb{N} \text{ and } |x-9| > |x|\}$ (no idea how to get started on this one!)

4. $4\mathbb{Z} \times 2\mathbb{Z}$

for 1 I think it is finite since the set of integers, $\mathbb{Z}$ is countable

for 2 I believe it is uncountable since $\mathbb{R}$ itself is uncountable

for 3 I have no idea where to begin

for 4 I guess this is the cross product of the set of Integers, not sure where to start for this either.

Answer 1

1. Since $\Bbb Z$ and countable and there is a bijection between this set and $\Bbb Z$, this set is countable too.
2. By that argument, $\{0\}$ is uncountable, since it is $\Bbb R\setminus(\Bbb R\setminus\{0\})$. The set $\Bbb R\setminus\Bbb N$ is uncountable since it is the complement of a countable set in an uncountable set.
3. That set is finite, since it is equal to $\{1,2,3,4\}$.
4. It's the Cartesian product of two countable sets, and therefore it's countable.

Answer 2

Hints (no answers)

1. Consider $f: \mathbb Z\setminus\{0\} \to \{1/m | m ∈ \mathbb{Z}, m ≠ 0\}$ where $f(m) = \frac 1m$.

Is that a bijection? If so $| \{1/m | m ∈ \mathbb{Z}, m ≠ 0\}| = |\mathbb Z\setminus\{0\}|$.

2. You need to call an a theorem about what the cardinality of removing the a countable elements from an uncountable set. Do you know the theorem I am referring to.

3. Well, what is $M = \{x | x ∈ \mathbb{N} \text{ and } |x-9| > |x|\}$

If $n = 59$ then $|n-9|= |59- 9| = |50| = 50$ and $|n|=|59| = 59$ and $|n-9|=50 < 59=|n|$ so $59\not \in M$. And if $n = 3$ then $|n-9| = |3-9| =|-6| = $ and $|n|=|3| = 3$ and $|n-9| = 6>3 = |n|$ so $3 \in M$.

And $\pi \not \in \mathbb N$ so $\pi \not \in M$. So what IS $M$ if we know $59$ is not in $M$, and $3$ is in $M$ and $\pi$ and $\text{Babar, the elephant}$ are not in $M$. What is and is not in $M$.

4. What is the cardinality of $4\mathbb Z$? Of $2\mathbb Z$. What do you know about that cardinalities of Cartesian products?

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More on 3:

$|x-9| = x-9$ if $x- 9 \ge 0$ and $|x-9| = 9-x$ if $x -9 < 0$. And $|x|=x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$.

So you have four cases

1. If $x -9 \ge 0$ and $x \ge 0$ then if $x-9 > x$ we have $x \in M$.

2. If $x - 9 < 0$ and $x \ge 0$ then if $9-x > x$ we have $x\in M$.

3. If $x-9 \ge 0$ and $x < 0$ then if $x-9 > - x$ we have $x \in M$.

4. If $x-9 < 0$ and $x < 0$ then if $9-x > -x$ we have $x\in M$.

those can be restated as

1. If $x \ge 9$ and $x \ge 0$ then $-9 > 0$ then $x \in M$.

2. If $x \le 0 < 9$ and $2x > 9$ then $x \in M$.

3. If $x < 0 < 9 \le x$ then if $2x > 9$ then $x \in M$.

4. If $x< 9$ and $x < 0$ then if $9 > 0$ then $x \in M$.

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Note only one of those is possible.