Question

3. (20 pt.) Prove that the following language is not regular using the closure properties of regular languages. C = {0"1"|m,n0 and mon} Hint: find a regular language L such that CNL is not regular and use the closure properties of regular languages to show that this means that C is not regular.

Answer 1

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Stept - Language Given is C=rom "/m, n20 and mens V When we break this Language into 2 parts VL C = {oma/m, n2o and man} U coo) rom," /mn>o and m<n Let L = { om n/m, nzo and m>n} and 12 = Yom_m/m,nzo and many c=4 ULL Step2 4 = {omrlman} V Non-Finite . Non Regular CFLC context free language) because 0001111 xxx | skip infinite times Stack So, this is not Regular this is [FL Similiardy L is also CFL

d Steb3 = Now, according to Closure boobooties of Languages c=4 ULZ CFL UCFL=CFL Hence, we can say that CFL is not regular
Refer this closure properties table

Closure Properties REG DCFL RE CFL ✓ CSL ✓ REC ✓ ✓ X ✓ X ✓ ✓ X ✓ X ✓ х X ✓ Operations Union Intersection Complement Concatenation Kleene star Homomorphism Inverse Homomorphism Reverse Substitution (E - free) ✓ ✓ ✓ X ✓ ✓ ✓ ✓ ✓ ✓ х X X ✓ ✓ ✓ х ✓ ✓ ✓ X ✓ ✓