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Regular languages are not closed under infinite union because while the union of a finite number of regular languages results in a regular language, an infinite union can produce a language that is not regular. For example, the set of languages {a^n | n ≥ 0} for n = 0, 1, 2, ... represents an infinite union of regular languages, but the resulting language {a^n | n ≥ 0} is not regular, as it cannot be recognized by any finite automaton. This is due to the limitations of finite state machines, which cannot handle the potentially unbounded complexity of infinite unions.
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Is a rational number closed under addition?
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.
When are complex numbers closed under addition?
Quite simply, they are closed under addition. No "when".
Are polynomial expressions closed under subtraction?
Yes they are closed under multiplication, addition, and subtraction.
Are rational numbers closed under division multiplication addition or subtraction?
Rational numbers are closed under addition, subtraction, multiplication. They are not closed under division, since you can't divide by zero. However, rational numbers excluding the zero are closed under division.
What is a set of rational numbers not closed under?
It is not closed under taking square (or other even) roots.
How can you prove that the complement of a regular language is regular?
The complement of a regular language is regular because regular languages are closed under complementation. This means that if a language is regular, its complement is also regular.
Are decidable languages closed under concatenation?
Yes, decidable languages are closed under concatenation.
Are decidable languages closed under intersection?
Yes, decidable languages are closed under intersection.
Are recognizable languages closed under concatenation?
Yes, recognizable languages are closed under concatenation.
Is the class of recognizable languages closed under complementation?
No, the class of recognizable languages is not closed under complementation.
Is the class of undecidable languages closed under complementation?
No, the class of undecidable languages is not closed under complementation.
Is the set of nonregular languages closed under intersection?
No, the set of nonregular languages is not closed under intersection.
Are Turing recognizable languages closed under concatenation?
Yes, Turing recognizable languages are closed under concatenation.
Are Turing recognizable languages closed under intersection?
Yes, Turing recognizable languages are closed under intersection.
Are Turing recognizable languages closed under union?
Yes, Turing recognizable languages are closed under union.
Are context-free languages closed under concatenation?
Yes, context-free languages are closed under concatenation.
What are the closure properties of regular sets?
;: Th. Closed under union, concatenation, and Kleene closure. ;: Th. Closed under complementation: If L is regular, then is regular. ;: Th. Intersection: .
