Mathematics GU4041x
Introduction to Modern Algebra
Assignment #3
Due September 26, 2016
In what follows, you may assume anything stated in class except what is being asked: notably, you
may use the associative, commutative, and distributive properties of multiplication.
1. Recall that the three Peano axioms, as stated in class, were roughly: (P1) 0 is not a successor;
(P2) the successor function is injective; (P3) the axiom of induction. For each i= 1,2,3,
define a set Ni, an element 0i, and a function fi:Ni→Ni, and prove that it does not satisfy
the ith Peano axiom but does satisfy the other two. Are such sets unique in the same way
that Nis?
2. (Ex 2 from class) Prove that for all x, y, z ∈N,x+z=y+zimplies x=y.
3. (Ex 3 from class) Prove that for all x∈N,x < x0.
4. The well-ordering principle says that every nonempty subset of Ncontains a least element.
Prove that this element is unique.
5. Prove that Nhas no greatest element, that is, no element greater than or equal to every
element of N.
6. Prove that for all x, y, z ∈N,y≤zimplies xy ≤xz.
7. For x, y ∈N, propose a rule to define xysimilar to those for x+yand x·ygiven in class. (You
don’t have to prove that it exists, though it would follow the argument for x+ystraightfor-
wardly.)
8. Asequence of natural numbers is just a function a:N→N. Its values a(i) are frequently
denoted ai. For example, ai=i2+ 1. For such a sequence, we may define Pn
i=0 aiby the rule
P0
i=0 ai=a0,
Pn0
i=0 ai= (Pn
i=0 ai) + an0
and then use the axiom of induction to prove that this assigns an unique value to Pn
i=0 ai.
Assuming this, use the axiom of induction to prove that 2Pn
i=0 i=n2+n. (You may assume
the standard notation 1 = 00, 2 = 1 + 1 = 000,n2=n·n, and so on.)
