Design Turing machines $($TMs$)$ to perform the following tasks. In each case, ensure $[25$ marks$]$

you include a description of your algorithm.

$($a$)$ For a tape containing an integer k in unary form, construct a TM to replace its $[7$ marks$]$

input by f$($k$)=$k$+1$

$($b$)$ For a tape containing an integers i and j in unary form separated by a blank b$,[7$ marks$]$

construct a TM to replace its input by f$($i$,$j$)=$i$+$j

$($c$)$ Construct a TM to recognise inputs over the alphabet $\{0,1\}$ containing the $[11$ marks$]$

substring $101$