# 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.

**Answer:** **(B)** **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, a^{n} b^{n} for all n≥0 is non-regular as well as a^{n} b^{m} 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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