Let $ S \equal{}\{1,2,3, \ldots, 2n\}$ ($ n \in \mathbb{Z}^\plus{}$). Ddetermine the number of subsets $ T$ of $ S$ such that there are no 2 element in $ T$ $ a,b$ such that $ |a\minus{}b|\equal{}\{1,n\}$
1. **Define the set and the problem constraints:**
   Let \( S = \{1, 2, 3, \ldots, 2n\} \) where \( n \in \mathbb{Z}^+ \). We need to determine the number of subsets \( T \subseteq S \) such that there are no two elements \( a, b \in T \) with \( |a - b| = 1 \) or \( |a - b| = n \).

2. **Associate subsets to strings:**
   We will represent each valid subset \( T \) by an \( n \)-digit string with characters from the set \(\{N, F, S\}\):
   - \( N \) if neither \( i \) nor \( n+i \) are in the subset.
   - \( F \) if \( i \) is in the subset.
   - \( S \) if \( n+i \) is in the subset.
   
   This ensures that \( i \) and \( n+i \) cannot both be in the subset \( T \).

3. **Establish constraints on the strings:**
   We need to count the number of strings in \(\{N, F, S\}^n\) such that:
   - There are no two consecutive \( F \)'s.
   - There are no two consecutive \( S \)'s.
   - The string does not start with \( S \) and end with \( F \).

4. **Define the recursion:**
   Let \( a_n \) be the number of valid strings of length \( n \) without two consecutive \( F \)'s or \( S \)'s.
   Let \( b_n \) be the number of valid strings of length \( n \) that start with \( S \) and end with \( F \).

5. **Establish the recurrence relations:**
   - For \( a_n \):
     \[
     a_n = a_{n-1} + 2a_{n-2}
     \]
     This is because a valid string of length \( n \) can be formed by:
     - Adding \( N \) to a valid string of length \( n-1 \).
     - Adding \( F \) to a valid string of length \( n-1 \) that ends in \( N \).
     - Adding \( S \) to a valid string of length \( n-1 \) that ends in \( N \).

   - For \( b_n \):
     \[
     b_n = b_{n-1} + 2b_{n-2}
     \]
     This is because a valid string of length \( n \) that starts with \( S \) and ends with \( F \) can be formed by:
     - Adding \( N \) to a valid string of length \( n-1 \) that starts with \( S \) and ends with \( F \).
     - Adding \( F \) to a valid string of length \( n-1 \) that starts with \( S \) and ends in \( N \).
     - Adding \( S \) to a valid string of length \( n-1 \) that starts with \( S \) and ends in \( N \).

6. **Subtract to find the desired count:**
   The number of valid subsets \( T \) is given by:
   \[
   a_n - b_n
   \]

The final answer is \( \boxed{ a_n - b_n } \).