In each of the following eight cases, indicate whether the given statement is tr

Question

In each of the following eight cases, indicate whether the given statement is true
or false.

I have answered them all and I want to make sure they are correct.

(a) Let E be a set and suppose that there exists a surjective function f : R ---> E. Then
E is uncountable.

TRUE because R is uncountable

(b) If E is a subset of R which has a supremum, then the set -E = {-x : x in E} has
an infimum.

TRUE because E has a sup iff -E has an inf.

(c) Let a be in R. Then |a| <epsilon for all epsilon> 0 if and only if a = 0.

TRUE

(d) If {Ex}x in R is a collection of infinite sets indexed by the real numbers, then (the union) U x in R Ex
is at most countable.

FALSE

e) Every subset of R has at most two suprema.

FALSE

(f) Let f : R ---> R be defined by f(x) = x^2. Then f^-1([0, 1]) = [-1, 1].

FALSE

(g) Let A1,A2,A3,.... be nonempty finite subsets of N such that An intersect Am = nonempty set for all
distinct n,m in N. Define the function f : N --> N by declaring f(n) to be the least
element of An. Then f is injective.

FALSE

(h) Let A1,A2,A3,..... be nonempty bounded subsets of R such that An intersect Am = nonempty set for
all distinct n,m in N. Define the function f : N --> R by f(n) = supAn. Then f is
injective.

TRUE

Explanation / Answer

1. True 2. False 3. False 4. Fasle 5. True 6. True 7.False 8. True

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