Let X be a set with exactly n elements, where n ≥ 2, and consider the group P(X) with the symmetric difference operation A o B := AAB as in Coursework 1. (i) Show that there exist n elements 9₁,..., 9n such that (91,92,..., 9n) is equal to P(X). (You do not have to give a detailed proof that your reasoning is correct, but in your answer you should describe a list of n elements sufficient to generate the group.) (ii) Prove that there do not exist n - 1 elements h₁,..., hn-1 such that (h₁,..., hn-1)= P(X).
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The question you provided is related to concepts from abstract algebra and set theory. Here are the key concepts involved:
Set Theory: The question starts with the set X, which has n elements. Set theory deals with the properties and relationships of sets, including operations like union (∪), intersection (∩), and symmetric difference (∆).
Group: In mathematics, a group is a set equipped with a binary operation that satisfies certain properties, such as associativity, the existence of an identity element, and the existence of inverses. The group P(X) is defined as the power set of X with the symmetric difference operation (∆), and you need to demonstrate properties related to it.
Symmetric Difference: Symmetric difference is an operation on sets where A ∆ B (A symmetric difference B) is defined as the set of elements that are in A or in B, but not in their intersection.
Generators of a Group: In group theory, a set of elements is considered a generating set if it can be used to generate all elements of the group through repeated applications of the group operation.
Vector Space and Dimension: The concept of dimension, often used in linear algebra, is mentioned in the context of vector spaces. In this case, the group P(X) is considered as a vector space over the field with two elements (F_2). The dimension of a vector space is related to the number of linearly independent elements needed to generate the entire space.
In summary, the question involves elements of group theory, set theory, and vector spaces, with a focus on generating sets and dimensions within the context of the group P(X) and its operations.
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