MAT 300 Mathematical Structures / 100
Final Exam11:40class May 11, 2004 name
Minor type-ohs corrected in this published version
High quality precise ar-
guments are more impor-
tant than just quantity.
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You may use the following theorem without having to prove it:
Thm. 6.4.1. If a, b ∈Zare not both zero, then there exist x, y ∈Zsuch that gcd(a, b) = xa +yb.
1. Precisely state the definitions of FIVE of the following:
a. relation, b. function, c. one-to-one, d. power set, e. partition, f. divides, g. countable.
2. State ONE of the “Axiom of Induction”, or the “Principle of Mathematical Induction”.
3. Suppose f:A7→ Bis a function from a set Ato a set B,C⊆Aand D⊆Bare subsets, and
{Aα:α∈Λ}is an indexed collection of subsets of A. Write XCfor the complement of a subset X.
a. Show that f(Sα∈ΛAα)⊆Sα∈Λf(Aα).
b. Which of f(CC) = (f(C))Cand f−1(DC) = (f−1(D))Cis always true?
c. Give a counterexample to show that one of the statements in b. is not true for all A, B, C , D, f.
4. Suppose A,Band Care sets and g:A7→ Band f:B7→ Care functions.
a. Show that if both fand gare one-to-one then the composition f◦gis also one-to-one.
b. What can you conclude about for g(or both) if f◦gis one-to-one? Prove your result.
5. Recall that a function f:R7→ Ris called increasing if x1≤x2implies that f(x1)≤f(x2).
a. Show that if f, g:R7→ Rare both increasing, then f◦gis also increasing.
For n∈Z+recursively define f(n)by f1=f, and f(n+1) =f◦f(n).
b. Show by induction that if f:R7→ Ris increasing then f(n)is increasing for every n∈Z+.
6. a. Prove Thm 6.4.3: If a, b ∈Zare relatively prime, c∈Z, and a|bc, then a|c. (Hint: c=c·1.)
b. Give a counterexample (values for b, c) that shows that 15 |bc need not imply 15 |bor 15 |c.
c. Show that there does not exist any rational number xsuch that x2= 15. (Hint: For an indirect
proof, assume there exist integers p, q with gcd(p, q) = 1 such that p
q2= 15. Use part a.)
7. Define a relation ≺on Z+\ {1}by x≺yif and only if yis a multiple of x(written x|y).
a. Show that ≺is a partial order on Z+\ {1}.
b. Find all minimal elements in Z+\ {1}under the partial order ≺. Prove your result.
c. Find all smallest elements in Z+\ {1}under the partial order ≺.Bonus: Prove your result.
Bonus. Prove by induction that for all n∈Z+,5n≥n2.
Bonus. Use the well-ordering principle to prove: “Every fraction can be written in lowest terms”.
Bonus. Show that every nonzero element in Znhas a multiplicative inverse if and only if nis prime.