Let A be an infinite set and let B ⊆ A be a subset. Prove:
(a) Assume A has a denumerable subset, show that A is equivalent
to a proper subset of A.
(b) Show that if A is denumerable and B is infinite then B is
equivalent to A.
Let A and B be two non empty bounded subsets of R:
1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A
+ sup B
2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup
A
hint:( show c supA is a U.B for cA and show if l < csupA then
l is not U.B)
(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then
a∣c.
(B) Let p ≥ 2. Prove that if 2p−1 is prime, then p
must also be prime.
(Abstract Algebra)
Let V = { S, A, B, a, b, λ} and T = { a, b }, Find the
languages generated by the grammar G = ( V, T, S, P } when the set
of productions consists of:
S → AB, A → aba, B → bab.
S → AB, S → bA, A → bb, B → aa.
S → AB, S → AA, A → Ab, A → a, B → b.
S → A, S →...
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 be the sum of the last four digits and let B be the last
digit of your 8-digit student ID. (Example: For 20245347, A = 19
and B = 7) On a road trip, a driver achieved an average speed of
(48.0+A) km/h for the first 86.0 km and an average speed of
(43.0-B) km/h for the remaining 54.0 km. What was her average speed
(in km/h) for the entire trip? Round your final answer to three
significant...
Let f: A → B be a function, and let {Bi: i ∈ I} be a partition
of B. Prove that {f−1(Bi): i ∈ I} is a partition of A. If
~I is the equivalence relation corresponding to the
partition of B, describe the equivalence relation corresponding to
the partition of A. [REMARK: For any C ⊆ B, f−1C) = {x ∈ A: f(x) ∈
C}.]
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