Prove the following statements!
1. If A and B are sets then
(a) |A ∪ B| = |A| + |B| − |A ∩ B| and
(b) |A × B| = |A||B|.
2. If the function f : A→B is
(a) injective then |A| ≤ |B|.
(b) surjective then |A| ≥ |B|.
3. For each part below, there is a function f : R→R that is
(a) injective and surjective.
(b) injective but not surjective.
(c) surjective but not injective.
(d)...
Please Prove the following, be clear and percise.
By removing sets with ever decreasing length, show that we can
construct a "Cantor-like" set which has positive length. How large
can we make the length of this set?
Prove the following:
(a) Let A be a ring and B be a field. Let f : A → B be a
surjective homomorphism from A to B. Then ker(f) is a maximal
ideal.
(b) If A/J is a field, then J is a maximal ideal.
Let A = Z and let a, b ∈ A. Prove if the following binary
operations are (i) commutative, (2) if they are associative and (3)
if they have an identity (if the operations has an identity, give
the identity or show that the operation has no identity).
(a) (3 points) f(a, b) = a + b + 1
(b) (3 points) f(a, b) = a(b + 1)
(c) (3 points) f(a, b) = x2 + xy + y2
a.) Prove the following: Lemma. Let a and b be integers. If both
a and b have the form 4k+1 (where k is an integer), then ab also
has the form 4k+1.
b.)The lemma from part a generalizes two products of integers of
the form 4k+1. State and prove the generalized lemma.
c.) Prove that any natural number of the form 4k+3 has a prime
factor of the form 4k+3.