Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the…
2003
Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings
- A.
does not form a group
- B.
forms a non-commutative group
- C.
does not have a right identity element
- D.
forms a group if the empty string is removed from ∑*
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Correct answer: A
Answer: Σ* with concatenation does not form a group.
Reasoning:
Closure: For any strings x and y in Σ*, the concatenation xy is also a string in Σ*, so closure holds.
Associativity: String concatenation is associative, i.e., (xy)z = x(yz) for all strings x, y, z.
Identity: The empty string (denoted ε) is an identity element because for any string x, εx = x and xε = x.
Inverses: For a string x to have an inverse y we would need xy = yx = ε. Using string lengths, length(xy) = length(x) + length(y). This equals 0 only if both lengths are 0, so only the empty string ε has an inverse (itself). Any non-empty string cannot have an inverse under concatenation.
Conclusion: Because not every element has an inverse, Σ* with concatenation is not a group. It is, however, a monoid (associative with an identity).
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