Question 9231 – GATE 2007

If a set is finite then it must be regular , as every language which contains finite elements is regular. Hence, every finite subset of a non-regular set is regular.
Every subset of regular set is regular, is false. For example L = {an bn | n ≥ 0} is subset of ∑* and L is CFL, whereas ∑* is regular. Hence, every subset of regular set need not be regular.
The union of two non-regular sets is not regular, is also a false statement.
For example, consider two CFL’s.
L = {aⁿ bⁿ | n ≥ 0} and its complement L^c = {a^m bⁿ | m ≠ n } U b*a*.
If we take UNION of L and L^c , we will get ∑*, which is regular. Hence the UNION of two non-regular set may or may not be regular.
The statement, Infinite union of finite sets is regular is also a false statement.