enthusiast2 - Numbers and Sets exercises for enthusiasts 2...

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Unformatted text preview: Numbers and Sets - exercises for enthusiasts 2. W. T. G. 1. Does there exist an uncountable family B of subsets of N such that for every A,B ∈ B (distinct) the intersection of A with B is finite? 2. Is it possible to write the closed interval [0 , 1] as a countably infinite union of disjoint closed (non-empty) subintervals? 3. Let f be a function from R 2 to R such that, for every x and y in R , the functions z 7→ f ( x,z ) and w 7→ f ( w,y ) are polynomials. Prove that f is a polynomial in x and y . Does the result hold if R is replaced by Q ? 4. Let X be a non-empty set with an associative binary operation defined on it (which is written as multiplication in what follows). Suppose that for every x ∈ X there is a unique x * such that xx * x = x . Prove that X is a group. 5. Is there a field that can be made into an ordered field in exactly three ways? 6. Given n points in the plane, not all collinear, show that it is possible to find a line containing exactly two of them....
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