a,b]
 [a,b)
 (−∞,b)
 [a,+∞)
 (a,b)
 (−∞,+∞)
 (−∞,b]
 (a,+∞)
 (a,b]

    {x∣a≤x}

{x∣x≤b}
{x∣x<b}
{x∣x∈R}
{x∣a<x<b}
{x∣a<x}
{x∣a≤x≤b}
{x∣a<x≤b}
{x∣a≤x<b}

 

It looks like you are providing a list of different types of intervals and set notations for representing real numbers. Here's a breakdown of each notation and how it relates to the concept of intervals on the real number line:

### Intervals:
1. **[a,b]**: Closed interval from \(a\) to \(b\). Includes both \(a\) and \(b\). 
   - Set notation: \(\{x \mid a \leq x \leq b\}\)
   
2. **[a,b)**: Half-open interval from \(a\) to \(b\). Includes \(a\) but not \(b\).
   - Set notation: \(\{x \mid a \leq x < b\}\)
   
3. **\((-∞,b)\)**: Open interval from \(-∞\) to \(b\). Includes all numbers less than \(b\) but not \(b\).
   - Set notation: \(\{x \mid x < b\}\)

4. **[a,+∞)**: Half-open interval from \(a\) to \(+\infty\). Includes all numbers greater than or equal to \(a\).
   - Set notation: \(\{x \mid a \leq x\}\)

5. **(a,b)**: Open interval from \(a\) to \(b\). Includes all numbers between \(a\) and \(b\), but excludes both \(a\) and \(b\).
   - Set notation: \(\{x \mid a < x < b\}\)

6. **\((-∞,+∞)\)**: The set of all real numbers (open on both ends).
   - Set notation: \(\{x \mid x \in \mathbb{R}\}\)

7. **\((-∞,b]\)**: Half-closed interval from \(-∞\) to \(b\). Includes all numbers less than or equal to \(b\).
   - Set notation: \(\{x \mid x \leq b\}\)

8. **(a,+∞)**: Open interval from \(a\) to \(+\infty\). Includes all numbers greater than \(a\).
   - Set notation: \(\{x \mid a < x\}\)

9. **(a,b]**: Half-open interval from \(a\) to \(b\). Includes \(b\), but excludes \(a\).
   - Set notation: \(\{x \mid a < x \leq b\}\)

### Set Notations:
- \(\{x \mid a \leq x\}\): All numbers \(x\) such that \(x\) is greater than or equal to \(a\).
- \(\{x \mid x \leq b\}\): All numbers \(x\) such that \(x\) is less than or equal to \(b\).
- \(\{x \mid x < b\}\): All numbers \(x\) such that \(x\) is less than \(b\).
- \(\{x \mid x \in \mathbb{R}\}\): All real numbers (i.e., the entire real number line).
- \(\{x \mid a < x < b\}\): All numbers \(x\) such that \(x\) is strictly between \(a\) and \(b\).
- \(\{x \mid a < x\}\): All numbers \(x\) such that \(x\) is greater than \(a\).
- \(\{x \mid a \leq x \leq b\}\): All numbers \(x\) such that \(x\) is between \(a\) and \(b\), including both \(a\) and \(b\).
- \(\{x \mid a < x \leq b\}\): All numbers \(x\) such that \(x\) is between \(a\) and \(b\), excluding \(a\) but including \(b\).
- \(\{x \mid a \leq x < b\}\): All numbers \(x\) such that \(x\) is between \(a\) and \(b\), including \(a\) but excluding \(b\).

Would you like further details or examples? 

Here are 5 questions to explore:
1. How do open and closed intervals impact limits in calculus?
2. How can these interval notations be applied to inequalities?
3. What are the applications of intervals in domain and range of functions?
4. How do intervals relate to the continuity of functions?
5. Can you explain the difference between finite and infinite intervals?

**Tip:** Always visualize interval notation on the number line to better understand which values are included and excluded.

The query submitted by the user: [a,b], [a,b), (−∞,b), [a,+∞), (a,b), (−∞,+∞), (−∞,b], (a,+∞), (a,b] {x∣a≤x} {x∣x≤b} {x∣x<b} {x∣x∈R} {x∣a<x<b} {x∣a<x} {x∣a≤x≤b} {x∣a<x≤b} {x∣a≤x<b}

Explore different types of interval notations and set theory for representing real numbers, including closed and open intervals, infinite sets, and inequalities. Suitable for high school students studying advanced algebra and real number concepts.

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