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Let $B = \{ 123 \}$.

Note that $B$ is finite.

Let $L = \left \{ \left\langle M \right\rangle | M \text{ is a Turing machine such that } B \subseteq L(M) \right\}$.

Is it sufficient to show that $L$ is recognizable by checking if $M$ accepts on input $123$, and therefore $\left\langle M \right\rangle \in L$?

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Yes, it is sufficient.

$\langle M \rangle \in L \Longleftrightarrow B\subseteq L(M) \Longleftrightarrow M\text{ accepts all strings in }B \Longleftrightarrow M\text{ accepts }123$.

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