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Help with a Proof about 2D Arrays

Let $A[m][n]$ be the set of all $2\times 2$ arrays in which every entry is a square and 0 or 1. Prove that $A$ is not a permutation of $\{0, 1\}^{2^{2^3}}$.
So the array can only be 0 or 1 for each row. I think we need to show that the complement of a square array is a different array, or prove that given any array A that is not a square array, you can show that $\forall i, j \, A[i][j] eq A[j][i]$.

A:

Proof: If $A$ is a permutation of $\{0,1\}^{2^{2^3}}$, then since $A$ is a function $\{0,1\}^{2^{2^3}}\to\{0,1\}^{2^{2^3}}$, $A$ is one-to-one, and since \$|\{0,1\}^{2^{2^3}}| = 2^{2^{3+1

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