Math 415, Test 2, November 19, 2003
Instructions: Do all five of the following problems. Please do your best, and show all
appropriate details in your solutions.
1. Suppose f:A→Bis a function.
(a) Define what is meant by the pre-image f∗(V).
(b) Define what is meant by the image f∗(U).
(c) For a collection of subsets {Vj}j∈Jof B, prove that f∗\
j∈J
Vj=\
j∈J
f∗(Vj).
(d) Is it true that f∗(U1∩U2) = f∗(U1)∩f∗(U2)? Prove or provide a counterexample.
2. (a) Define the terms injection, surjection and bijection.
(b) Suppose that both f:A→Cand g:B→Dare bijections. Show that the function
h:A×B→C×Ddefined by h(a, b) = (f(a), g(b)) is a bijection.
(c) Find the inverse function of f: (−∞,0] →[1,∞) of f(x) = x2+ 1. Be sure to state the
domain and range of the inverse function, and to verify it is the inverse function of f.
3. (a) Define what is meant by an equivalence relation on a set A.
(b) Define the relation ∼on R2by (a, b)∼(c, d) iff a2+b2=c2+d2. Is ∼an equivalence
relation? Verify your assertion.
(c) Do the relation classes from the relation in (b) form a partition of R2? If not, explain
which properties of a partition are violated. If so, describe the partition, and explain why it is
a partition.
4. (a) Find a bijection from the set S={1,4,9,16,25, . . .}of all squares of natural numbers
onto Z.
(b) Find a bijection from (0,1) →(a, b) where a < b.
(c) Explain carefully why the irrational numbers in any interval (a, b) with a < b are uncount-
able.
5. (a) Draw a lattice diagram for (P(A),⊆) where A={a, b, c, d}.
(b) Explain why (P(A),⊆) is not a totally ordered set.
(c) Let (S, ≤) be a partially ordered set. What properties must ≤have? Be specific.
(d) Let (S, ≤) be any partially ordered set. Prove that if Shas a greatest element, then it is
unique.
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