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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?

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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.

For each of the following sets A, prove that |A| = |[0, 1]| by defining explicit function(s) between A and [0,1]. State the relevant properties that your function(s) satisfy (one-to-one, onto, or bijective), but you do not need to prove or justify these properties.

This problem involves constructing bijections between various sets A and the interval [0, 1], proving that |A| = |[0, 1]|. Functions include linear transformations, piecewise mappings, and the arctangent function. Suitable for advanced undergraduate students.

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