Calculate Absolute Value of K from Set Cardinality Sum

To find the value of $|k|$, we analyze the set $S = \{1, 2, 3, \dots, 12\}$ and the set $X$ of non-empty subsets $A \subseteq S$ satisfying the condition: $\min(A) + \max(A) = 13$

The sets $Y$ and $Z$ form a partition of the collection of subsets $A \in X$ that contain the element $1$ (separated by parity conditions on the subset elements/sum). Thus, the total count $n(Y) + n(Z)$ equals the number of elements in $X$ that contain $1$:

$n(Y) + n(Z) = n(\{A \in X : 1 \in A\})$

Let us determine $n(\{A \in X : 1 \in A\})$ step-by-step:

1. Determine $\min(A)$ and $\max(A)$: Since $1 \in A$ and all elements of $S$ are positive integers, the minimum element of $A$ must be: $\min(A) = 1$

2. Determine $\max(A)$: Using the given condition $\min(A) + \max(A) = 13$: $1 + \max(A) = 13 \implies \max(A) = 12$

3. Count the remaining choices: Any valid subset $A \in X$ containing $1$ must include $1$ and $12$. The remaining elements of $A$ must be chosen from the set of intermediate integers: $\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$

   This intermediate set contains $12 - 2 = 10$ elements.

4. Calculate $n(Y) + n(Z)$: Since each of the $10$ elements can either be included in or excluded from $A$ independently, the total number of such subsets $A$ is: $n(\{A \in X : 1 \in A\}) = 2^{10} = 1024$

Therefore, we have: $n(Y) + n(Z) = 1024$

Given that $n(Y) + n(Z) = k^2$: $k^2 = 1024$

Taking the absolute value: $|k| = \sqrt{1024} = 32$

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