The problem asks to define explicit functions between sets $A$ and the interval $[0, 1]$ and show that $|A| = |[0,1]|$. This means proving that the sets $A$ and $[0, 1]$ have the same cardinality by constructing appropriate bijections between them.

Let's break it down case by case:

(a) $A = [-1, 2]$

We need a function $f: [-1, 2] \to [0, 1]$. A simple linear function that maps endpoints correctly is: $f(x) = \frac{x + 1}{3}$

• This function is bijective because it is one-to-one, onto, and continuous.
• It maps $-1 \mapsto 0$ and $2 \mapsto 1$.

(b) $A = [1, 2] \cup [3, 4]$

For this set, you can split the domain into two intervals and map them linearly to $[0, 1]$. We define two piecewise functions for each subinterval: [ f(x) = \begin{cases} \frac{x - 1}{2}, & \text{if } x \in [1, 2], \ \frac{x - 3}{2} + \frac{1}{2}, & \text{if } x \in [3, 4]. \end{cases} ]

• The first part maps $[1, 2] \to [0, 0.5]$, and the second part maps $[3, 4] \to [0.5, 1]$.
• This is a bijective mapping.

(c) $A = (0, 1)$

The set $A = (0, 1)$ is already open and can be mapped trivially onto $(0, 1) \subseteq [0,1]$ using the identity function: $f(x) = x$

• This is already a bijection from $(0, 1) \to (0, 1)$.

(d) $A = \mathbb{R}$

For the real line $\mathbb{R}$, the set has the same cardinality as $[0, 1]$. A well-known bijection between $\mathbb{R}$ and $[0, 1]$ (or any interval) can be constructed using the tangent function, for instance: $f(x) = \frac{1}{2} + \frac{1}{\pi} \arctan(x)$

• This function is bijective, mapping $\mathbb{R} \to (0, 1)$.

In each case, the functions provided are continuous and bijective, thus proving that $|A| = |[0,1]|$ for each set.

Do you want further details or have any questions on any specific part?

Here are 5 related questions to consider:

1. How can we generalize this mapping process to arbitrary intervals?
2. What are some other examples of bijections between $\mathbb{R}$ and subsets of $[0, 1]$?
3. Can you construct a bijection between $[0,1]$ and $\mathbb{Q} \cap [0,1]$?
4. How would these mappings change if we were working with $(0,1]$ or $[0,1)$?
5. How does the choice of mapping function affect properties like continuity or differentiability?

Tip: In many cases, linear or trigonometric functions are effective for constructing bijections between real intervals.