GATE | GATE-CS-2007 | Question 7

Which of the following is TRUE?
(A) Every subset of a regular set is regular.
(B) Every finite subset of a non-regular set is regular.
(C) The union of two non-regular sets is not regular.
(D) Infinite union of finite sets is regular.

Explanation:
Some points for Regular Sets:

• A set is always regular if it is finite.
• A set is always regular if a DFA/NFA can be drawn for it.

Option A: Every subset of a regular set is regular is False.
For input alphabets a and b, a*b* is regular. A DFA can be drawn for a*b* but a n b n for n≥0 which is a subset of a*b* is not regular as we cannot define a DFA for it.

Option B: Every finite subset of a non-regular set is regular is True.
Each and every set which is finite can have a well-defined DFA for it so whether it is a subset of a regular set or non-regular set it is always regular.

Option C: The union of two non-regular sets is not regular is False.
For input alphabets a and b, an bn for all n≥0 is non-regular as well as an bm for n≠m is also non- regular but their union is a*b* which is regular.

Option D: TInfinite union of finite sets is regular is False.
For input alphabets a and b sets {ab}, {aabb}, {aaabbb}…….. are regular but their union {ab} U {aabb} U {aaabbb} U …………………….. gives {a n b n for n>0} which is not regular.

This solution is contributed by Yashika Arora.

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