Theory of Computation and Compilers MCQ - Context free languages

36 Which of the following statements are true?
I. The set of all odd integers is a monoid under multiplication.
II. The set of all complex number is a group under multiplication
III. The set of all integers under the operation * given by a * b = a+b-ab is a monoid
IV. Zs under symmetric difference defined by AB = (A-B) ∪ (B-A) is an abelian group

37 The grammars G = ( { s }, { 0, 1 }, p , s)
where p = (s —> 0S1, S —> OS, S —> S1, S —>0} is a

38 The logic of pumping lemma is a good example of

The pigeon hole principle is nothing more than the obvious remark: if you have fewer pigeon holes than pigeons and you put every pigeon in a pigeon hole, then there must result at least one pigeon hole with more than one pigeon. It is surprising how useful this can be as a proof strategy.
In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language. The proofs of these lemmas typically require counting arguments such as the pigeonhole principle. So the answer is 'A'

39 The intersection of CFL and regular language

40 For two regular languages
L₁ = (a + b)* a and L₂ = b (a + b ) *.
The intersection of L1 and L2 is given by