Homework Statement
Show that 2\mathbb{Z} + 5\mathbb{Z} = \mathbb{Z}
Homework Equations
where 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = Z
The Attempt at a Solution
For any n in Z, we can write
n= (5-4)n = 5n +(-4)n = 5n + 2(-2n)
And since 5n is in 5Z and 2(-2n) is in 2Z, we can form Z from...
Homework Statement
show
f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0
f(x)=\left\{0 \\\ x\leq 0
is differentiable everywhere, and show its derivative is continuous
Homework Equations
Product Rule and Chain Rule for derivatives. Definition of a derivative
f^{'}(a)=\frac{f(x)-f(a)}{x-a}...
Homework Statement
find the limsup and liminf of a_{n}=(-1)^{n}\left( 1+\frac{1}{n} \right)
Homework Equations
Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup is the limit of
a_{n}=sup_{(k>=n)}a_{k}
The...
Homework Statement
4a) If x, y are in R, prove that (R, +) acts on R2 by (x,y)*r = (x+r, y) for all (x,y) in R2 and for all r in R.
b) If (x,y) are in R2, find the orbit of (x,y). Describe geometrically.
Homework Equations
none that I can think of
The Attempt at a Solution
The...
Homework Statement
1 a) Prove that the group (nZ, +) acts on Z by a*g = a + g for all g in nZ and for all a in Z.
b)What are the orbits?
c)How many orbits are there? Do the set of orbits remind you of anything in number theory?
Homework Equations
not sure
The Attempt at a Solution...
Homework Statement
5.4: If sigma in S_n has cycle type n_1,...,n_r, what is sgn(sig)? (sgn is the sign homomorphism)
Homework Equations
sgn(sigma) = 1 if sigma is even. sgn(sigma) = -1 is sigma is odd
cycle type is the length of the cycle type. If n_2 = 2, sigma has two 2-cycles.
The...
Homework Statement
5.1: Prove that S_n is generated by the set {(1 2), (3 4),...,(n-1 n)}
Homework Equations
None that I know of
The Attempt at a Solution
Any element in S_n can be written as a product of disjoint n-cycles. So now I need to show any n-cycle can be written as a product of...