$1.$ Strings represented by the given regular expressions

a$.-?[0-9]+(\backslash 10^)?[1-9]*$

Valid examples:

$"123"($integer$)$

$"-45"($negative integer$)$

$"007^8"($integer with a caret symbol$)$

$"-99^101"($negative integer with a caret symbol$)$

Invalid examples:

$"0^5"($contains a zero after the caret$)$

$"10^^3"($multiple carets$)$

$"0"($requires at least one non$-$zero digit$)$

Explanation:

This regular expression represents numbers with optional negative signs and an optional caret symbol $(^)$ followed by one or more digits $(0-9)$ and zero or more non$-$zero digits.

b$.[$A$-$Z$]+$ and $([$A$-$Z$]+...)$

Valid examples for the first part:

"HELLO" $($all uppercase letters$)$

"ABC" $($uppercase letters$)$

$"$X$"($a single uppercase letter$)$

Valid examples for the second part $($in parentheses$)$:

"ABC..." $($uppercase letters followed by ellipsis$)$

$"$XYZ$......"($uppercase letters followed by multiple ellipsis$)$

Invalid examples:

$"123"($no uppercase letters$)$

"AbC" $($contains lowercase letters$)$

Explanation:

This regular expression matches sequences of one or more uppercase letters $($A$-$Z$),$ and the second part $($inside parentheses$)$ matches sequences of uppercase letters followed by ellipsis $($e$.$g$.,$ ABC...$).$

Step $2$

$2.$ Regular expressions for the given descriptions:

a$.$ Identifiers that start with an underscore and end with a numeric digit, with any number of alphanumeric characters in between:

Regular Expression:

$\_([$a$-$zA$-$Z$0-9]*[0-9])$

Valid examples:

$"\_$variable$1"($starts with underscore, ends with a digit$)$

$"\_$test$123"($starts with underscore, ends with a digit$)$

$"\_$x$9"($starts with underscore, ends with a digit$)$

Invalid examples:

$"123$variable" $($doesn$\text{'}$t start with an underscore$)$

"invalid" $($contains non$-$alphanumeric characters$)$

Explanation:

This regular expression matches identifiers that start with an underscore $(\_)$ and end with a numeric digit $(0-9).$ In between, it can have any number of alphanumeric characters $($letters or digits$).$

Step $3$

b$.$ Phone numbers with formats $(888)888-8888$ or $888-888-8888$:

Regular Expression:

$(\backslash (\backslash$d$\{3\}\backslash )\backslash$s$?|\backslash$d$\{3\}-)\backslash$d$\{3\}-\backslash$d$\{4\}$

Valid examples:

$"(555)555-5555"($formatted with parentheses$)$

$"123-456-7890"($formatted with dashes$)$

$"999-888-7777"($formatted with dashes$)$

Invalid examples:

$"555-5555"($missing part of the number$)$

$"(123)-456-7890"($incorrect parentheses placement$)$

$"5555555555"($incorrect spacing$)$

Explanation:

This regular expression matches phone numbers in two formats $-$ either with parentheses and spaces or with dashes. It matches three digits, followed by either parentheses and optional spaces or dashes, and then three more digits, another dash, and four more digits.

c$.$ The VCU V Number:

Regular Expression:

VCU$\backslash$s$?$V$\backslash$s$?$Number

Valid example:

"VCU V Number" $($matches the specified format$)$

Invalid examples:

"VCU Number" $($missing $"$V$"$ between "VCU" and "Number"$)$

"VCUVNumber" $($no spaces as per the pattern$)$

Explanation:

This regular expression matches the specific pattern "VCU V Number" with optional spaces between "VCU," $"$V$,"$ and "Number."

Answer

In conclusion, we've discussed and provided solutions with examples for two sets of regular expressions and three given descriptions.

For the first set of regular expressions:

$1.$ The first regular expression represented strings that could be integers, possibly negative, and may include a caret symbol $(^)$ with specific rules for valid and invalid cases.

$2.$ The second regular expression represented strings containing uppercase letters, where the second part allowed for sequences of uppercase letters followed by ellipsis $(...).$

For the second set of regular expressions:

$1.$ We created a regular expression to match identifiers in a programming language. These identifiers must start with an underscore, end with a numeric digit, and may contain any number of alphanumeric characters in between.

$2.$ We designed a regular expression to match phone numbers in either $(888)888-8888$ or $888-888-8888$ formats.

$1.$ Strings represented by the given regular expressions:

a$.-?[0-9]+(\backslash 10^)?[1-9]*$

Valid examples:

$"123"($integer$)$

$"-45"($negative integer$)$

$"007^8"($integer with a caret symbol$)$

$"-99^101"($negative integer with a caret symbol$)$

Invalid examples:

$"0^5"($contains a zero after the caret$)$

$"10^^3"($multiple carets$)$

$"0"($requires at least one non$-$zero digit$)$

Explanation:

This regular expression represents numbers with optional negative signs and an optional caret symbol $(^)$ followed by one or more digits $(0-9)$ and zero or more non$-$zero digits.

b$.[$A$-$Z$]+$ and $([$A$-$Z$]+...)$

Valid examples for the first part:

"HELLO" $($all uppercase letters$)$

"ABC" $($uppercase letters$)$

$"$X$"($a single uppercase letter$)$

Valid examples for the second part $($in parentheses$)$:

"ABC..." $($uppercase letters followed by ellipsis$)$

$"$XYZ$......"($uppercase letters followed by multiple ellipsis$)$

Invalid examples:

$"123"($no uppercase letters$)$

"AbC" $($contains lowercase l

Each regular expression was accompanied by valid and invalid examples to illustrate their functionality and how they align with the provided descriptions. These examples help clarify the expected behavior of each regular expression and demonstrate their usage in real$-$world scenarios.