Final answer:
A student needs to prove that a language Y is not regular using the Pumping Lemma. By assuming Y is regular and applying the lemma, we reach a contradiction that violates Y's definition by either repeating sequences or breaking the pattern, thereby proving Y is not regular.
Step-by-step explanation:
The question asks to prove that a particular language Y over the alphabet Σ={1,#} is not regular. The set Y is defined as a collection of strings where each string w can be represented as w=x₁ #x₂ #⋅#⋅xk with k≥0, each xi ∈1*, and every xi is unique (i.e., xi is not equal to xj for i≠j). To prove that Y is not regular, one commonly used method is the application of the Pumping Lemma for regular languages.
To use the Pumping Lemma, we assume that Y is regular and then derive a contradiction. According to the lemma, there exists some integer p such that any string s in the language with length at least p can be divided into three parts, s=xyz, with certain properties that include the ability to 'pump' the middle portion y (i.e., repeat y any number of times) and still have the resulting string remain within the language.
Consider a string s=1p#1p#⋅#⋅#1p in Y. The Pumping Lemma would then suggest that s can be split into xyz such that xynz should be in Y for all n ≥ 0. However, repeating any segment of s that includes a portion of a '1' sequence or the '#' symbol would either result in a string with repeated xi sequences or violate the pattern of separation by '#', thus not satisfying the definition of Y. This leads to a contradiction, implying Y cannot be regular.