The action $$ a_{\alpha_0}\Phi_{0,3,6,10,13} = |0001001000100100\rangle, $$ can be obtained by subtracting the logical sum (AND operation) of \( \Phi_{0,3,6,10,13} \) and a word which represents only \( \alpha_0 \), that is $$ |1000000000000000\rangle, $$ from \( \Phi_{0,3,6,10,13}= |1001001000100100\rangle \).

This operation gives \( |0001001000100100\rangle \).

Similarly, we can form \( a^{\dagger}_{\alpha_4}a_{\alpha_0}\Phi_{0,3,6,10,13} \), say, by adding \( |0000100000000000\rangle \) to \( a_{\alpha_0}\Phi_{0,3,6,10,13} \), first checking that their logical sum is zero in order to make sure that the state \( \alpha_4 \) is not already occupied.