Question

Let a, b, and n be integers with n > 1 and (a, n) = d. Then

(i)First prove that the equation a·x=b has solutions in n if and only if d|b.

(ii) Next, prove that each of u, u+n′, u+ 2n′, . . . , u+ (d−1)n′ is a solution. Here,u is any particular solution guaranteed by (i), and n′=n/d.

(iii) Show that the solutions listed above are distinct.

(iv) Let v be any solution. Prove that v=u+kn′ for some k ∈ with 0≤k

Answer #1

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.

1. In this problem, the domain of x is integers. For each of the
statements, indicate whether it is TRUE or FALSE then write its
negation and simplify it to the point that no ¬ symbol occurs in
any of the statements (you may, however, use binary symbols such as
’̸=’ and <).
i. ∀x(x+ 2 ≠ x+3)
ii. ∃x(2x = 3x)
iii. ∃x(x^2 = x)
iv. ∀x(x^2 > 0)
v. ∃x(x^2 > 0)
2. Let A = {7,11,15}, B...

Let n ≥ 1 be an integer. Use the Pigeonhole Principle to prove
that in any set of n + 1 integers from {1, 2, . . . , 2n}, there
are two elements that are consecutive (i.e., differ by one).

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 A = {a,b,c,d}. Find an example of a relation on A that
is
(i) reﬂexive and symmetric.
(ii) not symmetric and not antisymmetric.
(iii) not symmetric but antisymmetric.
(iv) an equivalence relation
(v) a total order.

Let N* be the set of positive integers. The relation
∼ on N* is defined as follows: m ∼ n ⇐⇒ ∃k ∈
N* mn = k2
(a) Prove that ∼ is an equivalence relation.
(b) Find the equivalence classes of 2, 4, and 6.

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Let phi(n) = integers from 1 to (n-1) that are relatively prime
to n
1. Find phi(2^n)
2. Find phi(p^n)
3. Find phi(p•q) where p, q are distinct primes
4. Find phi(a•b) where a, b are relatively prime

Let B[1...n] be an array of integers. To express that no integer
occurs twice in the B We may write? (check all the answers that
apply)
a) forall i 1..n forall j in 1...n B[i] != B[j]
b)for all i in 1...n forall j in 1...n, i != j => B[i]
!=B[j]
c)forll i in 1...n forall j in 1...n i != j and B[i] != B[j]
d)forall i in 1...n forall j in 1...n B[i] =B[j] => i=j
e)forall...

Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...

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