Let L$1=\{$M $|$ M is a Turing machine that accepts the string $1\}.$

I.e$.,$ if the language accepted by M contains the string $1($possibly along with other strings$),$

then $$M $$ in L$1.$

Define the function f $($M$1,$ W$1)=$M$2$ as follows, where W$2$ denotes the input to M$2$:

$$ If $$M$1,$ W$1$ is not a valid encoding of a Turing machine M$1$ and an input string W$1,$

then make M$2$ reject.

$$ If $$M$1,$ W$1$ is a valid encoding, then make M$2$ simulate M$1$ on input W$1$ using a

universal Turing machine modified such that if M$1$ accepts W$1$ or M$1$ rejects W$1,$ then

M$2$ accepts W$2.$

That is$,$ f $($M$1,$ W$1)$ returns the encoding of a universal Turing machine M$2,$ where M$2$

simulates the execution of M$1$ on the input W$1,$ and accepts if M$1$ accepts or rejects W$1.$

$1.$ Give a brief argument to prove or disprove the following proposition:

f is computable.

$2.$ Give a formal argument to prove or disprove the following proposition:

$$M$1,$ W$1,$M$1,$ W$1$ in HALTT M $$ f $($M$1,$ W$1)$ in L$1.$

$3.$ Give a formal argument to prove or disprove the following proposition:

$$M$1,$ W$1,$M$1,$ W$1$ in HALTT M $$ f $($M$1,$ W$1)$ in L$1.$

$4.$ Give a brief argument to prove or disprove the following proposition:

L$1=\{1\}.$