Let n∈N, and let a1,a2,...an∈R. Prove that
|a1+a2+...+an|<or=|a1|+|a2|+...+|an|
Let n∈N, and let a1,a2,...an∈R. Prove that
|a1+a2+...+an|<or=|a1|+|a2|+...+|an|
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . ....
1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...
Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤...
Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...
Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) =...
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
1). Describe an algorithm that takes a list of n integers
a1,a2,...,an and find the average...
1). Describe an algorithm that takes a list of n integers
a1,a2,...,an and find the average of the largest and smallest
integers in the list.
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and...
1) Suppose a1, a2, a3, ... is a sequence of integers such that
a1 =1/16 and an = 4an−1. Guess a formula for an and prove that your
guess is correct.
2) Show that given 5 integer numbers, you can always find two of
the numbers whose difference will be a multiple of 4.
3) Four cats and five mice form a row. In how many ways can they
form the row if the mice are always together?
Please help...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪...
(4) Prove that, if A1, A2, ..., An are countable sets, then A1 ∪
A2 ∪ ... ∪ An is countable. (Hint: Induction.)
(6) Let F be the set of all functions from R to R. Show that |F|
> 2 ℵ0 . (Hint: Find an injective function from P(R) to F.)
(7) Let X = {1, 2, 3, 4}, Y = {5, 6, 7, 8}, T = {∅, {1}, {4},
{1, 4}, {1, 2, 3, 4}}, and S =...