The usual portable solution to implement an AND in case statements is to concatenate the boolean values:

case $A$B in
    11) echo "Both conditions are true";;
    1*) echo "Condition A is true";;
    *1) echo "Condition B is true";;
    00) echo "Both conditions are false";;
     *) echo "There is an unexpected error";;
esac

For your use case:

read -p "Enter a word: " word

A=0 B=0 C=0

case $word in    ( [aeiouAEIOU]* ) A=1;; esac
case $word in    ( *[0-9]        ) B=1;; esac
case $word in    ( ????          ) C=1;; esac

case $A$B$C in
  111)     echo "Four letters that start with a vowel and end with a digit" ;;
  11*)     echo "The word begins with a vowel AND ends with a digit."       ;;
  1* )     echo "The word begins with a vowel."                             ;;
  *1?)     echo "The word ends with a digit."                               ;;
   *1)     echo "The word is four letters long"                             ;;
    *)     echo "I don't understand what you've entered,"                   ;;
esac

Using a portable case for each boolean option. You can use ;;& in bash, or ;| in zsh. Sadly ksh doesn't have such option for case.

An alternative to set the booleans (in some shells: ksh, bash, zsh at least) is:

[[ $word ==   [aeiouAEIOU]* ]] && A=1 || A=0
[[ $word ==   *[0-9]        ]] && B=1 || B=0
[[ $word ==   ????          ]] && C=1 || c=0

The usual portable solution to implement an AND in case statements is to concatenate the boolean values:

case $A$B in
    11) echo "Both conditions are true";;
    1*) echo "Condition A is true";;
    *1) echo "Condition B is true";;
    00) echo "Both conditions are false";;
     *) echo "There is an unexpected error";;
esac

For your use case:

printf "Enter a word: "; read word

A=0 B=0 C=0

case $word in    ( [aeiouAEIOU]* ) A=1;; esac
case $word in    ( *[0-9]        ) B=1;; esac
case $word in    ( ????          ) C=1;; esac

case $A$B$C in
  111)     echo "Four letters that start with a vowel and end with a digit" ;;
  11*)     echo "The word begins with a vowel AND ends with a digit."       ;;
  1* )     echo "The word begins with a vowel."                             ;;
  *1?)     echo "The word ends with a digit."                               ;;
   *1)     echo "The word is four letters long"                             ;;
    *)     echo "I don't understand what you've entered,"                   ;;
esac

Using a portable case for each boolean option. You can use ;;& in bash, or ;| in zsh. Sadly ksh doesn't have such option for case.

An alternative to set the booleans (in some shells: ksh, bash, zsh at least) is:

[[ $word ==   [aeiouAEIOU]* ]] && A=1 || A=0
[[ $word ==   *[0-9]        ]] && B=1 || B=0
[[ $word ==   ????          ]] && C=1 || c=0

The usual solution to implement an AND in case statements is to concatenate the boolean values:

case $A$B in
    11) echo "Both conditions are true";;
    1*) echo "Condition A is true";;
    *1) echo "Condition B is true";;
    00) echo "Both conditions are false";;
     *) echo "There is an unexpected error";;
esac

For your use case:

read -p "Enter a word: " word

A=0 B=0 C=0

case $word in
     [aeiouAEIOU]*) A=1;;
     *[0-9])        B=1;;
    ????)          C=1;;
    *)             A=error; B=error; C=error;;
esac

case $A$B$C in
  111)     echo "Four letters that start with a vowel and end with a digit" ;;
  11*)     echo "The word begins with a vowel AND ends with a digit."       ;;
  1* )     echo "The word begins with a vowel."                             ;;
  *1?)     echo "The word ends with a digit."                               ;;
   *1)     echo "The word is four letters long"                             ;;
    *)     echo "I don't understand what you've entered,"                   ;;
esac

The usual portable solution to implement an AND in case statements is to concatenate the boolean values:

case $A$B in
    11) echo "Both conditions are true";;
    1*) echo "Condition A is true";;
    *1) echo "Condition B is true";;
    00) echo "Both conditions are false";;
     *) echo "There is an unexpected error";;
esac

For your use case:

read -p "Enter a word: " word

A=0 B=0 C=0

case $word in    ( [aeiouAEIOU]* ) A=1;; esac
case $word in    ( *[0-9]        ) B=1;; esac
case $word in    ( ????          ) C=1;; esac

case $A$B$C in
  111)     echo "Four letters that start with a vowel and end with a digit" ;;
  11*)     echo "The word begins with a vowel AND ends with a digit."       ;;
  1* )     echo "The word begins with a vowel."                             ;;
  *1?)     echo "The word ends with a digit."                               ;;
   *1)     echo "The word is four letters long"                             ;;
    *)     echo "I don't understand what you've entered,"                   ;;
esac

Using a portable case for each boolean option. You can use ;;& in bash, or ;| in zsh. Sadly ksh doesn't have such option for case.

An alternative to set the booleans (in some shells: ksh, bash, zsh at least) is:

[[ $word ==   [aeiouAEIOU]* ]] && A=1 || A=0
[[ $word ==   *[0-9]        ]] && B=1 || B=0
[[ $word ==   ????          ]] && C=1 || c=0