Econ 5100 Homework 2
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Econ 5100
Homework 2
Problem 1
We can view c as a binary relation between sets, because given any two sets A, B, either A c B or A c\ B . Let S be an arbitrary nonempty set and p(S) its power set. Is c a partial order on p(S)? Is it also a total order? Explain.
Problem 2.
Suppose R is a total order on S where |S| is finite. Prove that there exists a mapping f : S - R such that f(x) > f(y) if and only if xRy .
Problem 3.
Is d(x, y) = |x2 - y2 | a metric on R2? Explain.
Problem 4.
Define d : R2 - R+ such that d((x1 , x2 ), (y1 , y2 )) = max{|x1 - y1 |, |x2 - y2 |}. Show that d is a metric on R2 .
Problem 5.
Let (S, d) be a metric space and (xi ) be a sequence in S. Prove that if xi - x and xi - y, then x = y .
Problem 6.
Suppose xi - x and yi - y in (R, dE ). Prove that xi yi - xy .
Problem 7.
Given a metric space (S, d) , a sequence (xi ), and a point x e S, prove that if xi - x in (S, d), then the sequence di := d(xi , x) - 0 in (R, dE ).
Problem 8.
Prove that any intersection of finitely many open sets is open. (Hint: use induction.)
Problem 9.
Prove that given a metric space (S, d), T c S is closed in (S, d) if every conver- gent sequence (xi ) in T converges to some x e T.
Problem 10
Is {1} open, closed, both, or neither in (R, dE ) (Euclidean metric)? How about in (R, dD ) (discrete metric)? Explain.
2022-09-15