MAT 300 Mathematical Structures / 100
Test 2 November 3, 2006 name
Precision matters. Do not use any symbols that
have not been defined. Make sure that the logical
structure of your argument is absolutely clear.
1 2 3 4 5 Bonus
20 15 15 25 25 10
1. a. Precisely state the definitions of SIX of the following:
•Cartesian product •function •one-to-one •preimage
•reflexive •minimal •smallest •lower bound.
Start each definition by defining every symbol that you use, e.g.
“Suppose Sand Tare sets”, or “Suppose (W, ≤)is a partially ordered set”.
b. Give a precise statement of the Division algorithm.
2. Prove ONE of the following:
•Every integer greater than one is prime or can be factored into a product of primes.
•For every positive integer n, (n3+ 5n) is divisible by 6.
Bonus: Prove that for every integer n, (n3+ 5n) is divisible by 6.
3. Define a relation ∼on the set Z+×Z+of pairs of positive integers by (a, b)∼(c, d) if
and only if a+d=b+c. Show that ∼is an equivalence relation.
4. a. Suppose f:A7→ Bis a function from a set Ato a set B,
and {Bα:α∈Λ}is an indexed collection of subsets of B.
Show that f−1Ã[
α∈Λ
Bα!⊆[
α∈Λ
f−1(Bα),
b. Let Sc
Xdenote the complement of a subset S⊂Xin X. For EACH of the following decide
whether true or false. For ONE of the following, prove it, OR provide a counterexample.
•True. False. For every function f:A7→ Band every S⊆A,f(Sc
A)⊆(f(S))c
B.
•True. False. For every function f:A7→ Band every S⊆A,f(Sc
A)⊇(f(S))c
B.
•True. False. For every function f:A7→ Band every T⊆B,f−1(Tc
B)⊆(f−1(T))c
A.
•True. False. For every function f:A7→ Band every T⊆B,f−1(Tc
B)⊇(f−1(T))c
A.
5. Suppose A,Band Care sets and g:A7→ Band f:B7→ Care functions.
a. Suppose that both fand gare onto. Show that the composition f◦gis onto.
b. Suppose that f◦gis onto. What can you conclude about fand g?
Prove that you are correct.
Bonus. Define upper bound and least upper bound (for subsets of partially ordered sets, and
prove that if a subset of a partially ordered set has a least upper bound, then this is unique.