Summit: vague answers to well-known problems?

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Answers to Some of the Homework Problems

1. We just write out (i). Recall that X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2}. Hence x ∈ X1 =⇒ x ∈ X1 ∪ X2, so that X1 ⊆ X1 ∪ X2. By symmetry, X1 ⊆ X1 ∪ X2. Finally, suppose that Y is a set such that X1 ⊆ Y and X2 ⊆ Y . Let x ∈ X1 ∪X2. Then x ∈ X1 or x ∈ X2. If x ∈ X1, then x ∈ Y since X1 ⊆ Y . Similarly, if x ∈ X2, then x ∈ Y . Thus in either case x ∈ Y , so that X1 ∪X2 ⊆ Y . 2. (i) Injective, not su...

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5. Set h1 = (1, 2)(3, 4), h2 = (1, 3)(2, 4), h3 = (1, 4)(2, 3). Clearly h 2 i = 1 for all i, so that hi = h −1 i and thus (since 1 is its own inverse too) H is closed under taking inverses and contains 1. To see that H is closed in general, there is no problem with the products 1 · 1, 1 · hi = hi · 1, or hi , so we must just check hihj for i 6= j. For example, h1h2 = h3, and hence h2h1 = h −1 2...

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ژورنال

عنوان ژورنال: Nature

سال: 2002

ISSN: 0028-0836,1476-4687

DOI: 10.1038/418817a