Let a < c < b, and let f be defined on [a,b]. Show that f
∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover,
Integral a,b f = integral a,c f + integral c,b f .
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}.]