Question

3. Let N denote the nonnegative integers, and Z denote the integers. Define the function g : N→Z defined by g(k) = k/2 for even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a bijection.

(a) Prove that g is a function.

(b) Prove that g is an injection

. (c) Prove that g is a surjection.

Answer #1

1. A function f : Z → Z is defined by f(n) = 3n − 9.
(a) Determine f(C), where C is the set of odd integers.
(b) Determine f^−1 (D), where D = {6k : k ∈ Z}.
2. Two functions f : Z → Z and g : Z → Z are defined by f(n) =
2n^ 2+1 and g(n) = 1 − 2n. Find a formula for the function f ◦
g.
3. A function f :...

Let N denote the set of positive integers, and let x be a number
which does not belong to N. Give an explicit bijection f : N ∪ x →
N.

. Let n ∈ N. Prove (by induction) that n =
2knmn for some nonnegative kn ∈ Z
and some odd mn ∈ N. (Again, kn and mn may
depend on n.)

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Let n ∈ N and f : [n] → [n] a function. Prove that f is a
surjection if and only if f is an injection.

3. Let T be the set of integers that are not divisible
by 3. Prove that T is a countable set by finding a bijection
between the set T and the set of integers Z, which we know is
countable from class. (You need to prove that your function is a
bijection.)

Define the set E to be the set of even integers; that is,
E={x∈Z:x=2k, where k∈Z}. Define the set F to be the set of integers
that can be expressed as the sum of two odd numbers; that is,
F={y∈Z:y=a+b, where a=2k1+1 and
b=2k2+1}.Please prove E=F.

Due October 25. Let Z[i] denote the Gaussian integers, with norm
N(a + bi) = a 2 + b 2 . Recall that ±1, ±i are the only units i
Z[i]. (i) Use the norm N to show that 1 + i is irreducible in Z[i].
(ii) Write 2 as a product of distinct irreducible elements in
Z[i].

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

Prove that 1−2+ 2^2 −2^3 +···+(−1)^n 2^n =2^n+1(−1)^n+1 for all
nonnegative integers n.

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