2.1 Write regular expression for the following character sets, or reasons why no regular expression can be written: a. All strings of lowercase letters that begin and end in a. b. All strings of lowercase letters that either begin or end in a both) c. All strings of digits that contain no leading zeros d. All strings of digits that represent even numbers

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### Regular Expression Exercise

#### 2.1 Write a regular expression for the following character sets, or provide reasons why no regular expression can be written:

a. **All strings of lowercase letters that begin and end in `a`.**

b. **All strings of lowercase letters that either begin or end in `a` (or both).**

c. **All strings of digits that contain no leading zeros.**

d. **All strings of digits that represent even numbers.**

#### Explanation:

**a**: 
For strings that begin and end with `a`, the regular expression can be:
```
^a[a-z]*a$
```
Here, `^` asserts the position at the start of a line, `a` matches the start character, `[a-z]*` matches zero or more lowercase letters, and `a$` matches the last character and asserts position at the end of a line.

**b**: 
For strings that either begin or end with `a` (or both), the regular expression can be:
```
(^a[a-z]*$)|(^[a-z]*a$)
```
This combines two conditions: strings that begin with `a` or those that end with `a`.

**c**: 
For strings of digits that contain no leading zeros, the regular expression can be:
```
^[1-9][0-9]*$
```
Here, `^` asserts the position at the start of a line, `[1-9]` matches any digit from 1 to 9 (ensuring no leading zero), and `[0-9]*` matches zero or more digits.

**d**: 
For strings of digits that represent even numbers, the regular expression can be:
```
^[0-9]*[02468]$
```
This matches any sequence of digits ending in 0, 2, 4, 6, or 8, which represent even numbers.

### Notes:
- For part c, a string containing only "0" is not specified, but typically, "0" would be considered a special case.
- The regular expressions are written considering that `^` and `$` are used to assert the start and end of lines, respectively.
Transcribed Image Text:### Regular Expression Exercise #### 2.1 Write a regular expression for the following character sets, or provide reasons why no regular expression can be written: a. **All strings of lowercase letters that begin and end in `a`.** b. **All strings of lowercase letters that either begin or end in `a` (or both).** c. **All strings of digits that contain no leading zeros.** d. **All strings of digits that represent even numbers.** #### Explanation: **a**: For strings that begin and end with `a`, the regular expression can be: ``` ^a[a-z]*a$ ``` Here, `^` asserts the position at the start of a line, `a` matches the start character, `[a-z]*` matches zero or more lowercase letters, and `a$` matches the last character and asserts position at the end of a line. **b**: For strings that either begin or end with `a` (or both), the regular expression can be: ``` (^a[a-z]*$)|(^[a-z]*a$) ``` This combines two conditions: strings that begin with `a` or those that end with `a`. **c**: For strings of digits that contain no leading zeros, the regular expression can be: ``` ^[1-9][0-9]*$ ``` Here, `^` asserts the position at the start of a line, `[1-9]` matches any digit from 1 to 9 (ensuring no leading zero), and `[0-9]*` matches zero or more digits. **d**: For strings of digits that represent even numbers, the regular expression can be: ``` ^[0-9]*[02468]$ ``` This matches any sequence of digits ending in 0, 2, 4, 6, or 8, which represent even numbers. ### Notes: - For part c, a string containing only "0" is not specified, but typically, "0" would be considered a special case. - The regular expressions are written considering that `^` and `$` are used to assert the start and end of lines, respectively.
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