Due: Friday, February 16, 2007
Math 203 - Proving things (Algebra) / Problem Set 3
1) Prove the assertions made in the class:
a) The addition on
N
satisfies the cancellation law: ∀l , m, n ∈
N
one has: m+l=n+l⇐⇒ m=n.
b) The multiplication on
N
is associative, commutative, and distributive w.r.t the addition.
[Hint. Show first that (m+n)·l=m·l+n·lby induction on...]
2) Prove the assertion made in the class: The addition on
N
is compatible with the ordering, i.e., ∀l, m, n ∈
N
one has: m≤n⇐⇒ m+l≤n+l.
3) Prove that the ordering ≤on
N
is a total ordering, i.e., for all m, n ∈
N
one has: either m≤n, or n<m.
4) Let m, n ∈
N
satisfy m≤n. Then there exists a unique natural number lsuch that m+l=n(Why?).
We define the difference n−mof nand mby n−m:= l. Prove or disprove:
a) The difference is associative, commutative.
b) The difference is compatible with the ordering ≤.
c) The multiplication is distributive with respect to the difference.
5) Recall that for m, n ∈
N
we define: m|nDef
⇐⇒ m·l=nfor some l∈
N
.
a) Are there natural numbers nwhich are divisible by infinitely many m∈
N
?
b) Show that the divisibility |defines a partial ordering on
N
, which means by definition the following:
i) |is reflexive, i.e., n|nfor all n.
ii) |is anti-symmetric, i.e., m|n&n|m⇒m=n.
ii) |is transitive, i.e., l|m&m|n⇒l|n.
6) Let l, m, n ∈
N
, and suppose that n≥m. Show that the following are equivalent:
i) l|mand l|nand l|(n−m).
ii) ldivides two of the numbers m,n,n−m.
7) Let n > 1 be a natural number.
a) Show that there exits a prime number psuch that p|n.
b) Give all the details of Eucild’s proof showing that the set of all the prime numbers is not finite.
c) Show that ∀k∈
N
∃n∈
N
such that none of the numbers n, n + 1, . . . , n +kis a prime number.
This means that there are arbitrarily large sequences of consecutive non-prime numbers.
[Hint. For given l, find a number which is divisible by 2,3, . . . , l.]
8) Prove or disprove the following:
a) Every prime number p > 2 is of the form: p= 4 ·k±1 for some k∈
N
.
b) Every prime number p > 3 is of the form: p= 6 ·k±1 for some k∈
N
.
c) There exist infinitely many prime numbers of the form p= 4·k−1, p= 4 ·k+1, p= 6·k−1, p= 6 ·k+1.
