Defining the language PALINDROME, defined over Σ = {a,b}

Step 1:              a and b are in PALINDROME

Step 2:              if x is palindrome, then s(x)Rev(s) and xx will also be palindrome, where s belongs to Σ^*

Step 3:              No strings except those constructed in above, are allowed to be in palindrome

Defining the language {aⁿbⁿ }, n=1,2,3,… , of strings defined over Σ={a,b}

Step 1:              ab is in {aⁿbⁿ}

Step 2:              if x is in {aⁿbⁿ}, then axb is in {aⁿbⁿ}

Step 3:              No strings except those constructed in above, are allowed to be in {aⁿbⁿ}

Defining the language L, of strings ending in a , defined over  Σ={a,b}

Step 1:              a is in L

Step 2:              if x is in L then s(x) is also in L, where s belongs to Σ^*

Step 3:              No strings except those constructed in above, are allowed to be in L

Defining the language L, of strings beginning and ending in same letters , defined over  Σ={a, b}

Step 1:              a and b are in L

Step 2:              (a)s(a) and (b)s(b) are also in L, where s belongs to Σ^*

Step 3:              No strings except those constructed in above, are allowed to be in L

Defining the language L, of strings containing aa or bb , defined over       Σ={a, b}

Step 1:              aa and bb are in L

Step 2:              s(aa)s and s(bb)s are also in L, where s belongs to Σ^*

Step 3:              No strings except those constructed in above, are allowed to be in L

Defining the language L, of strings containing exactly one a, defined over       Σ={a, b}

Step 1:              a is in L

Step 2:              s(aa)s is also in L, where s belongs to b^*

Step 3:              No strings except those constructed in above, are allowed to be in L

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