(a) Prove that Sn is generated by the elements in the set {(i
i+1) : 1≤i≤n}.
[Hint: Consider conjugates, for example (2 3) (1 2) (2
3)−1.]
(b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for
n ≥ 3.
(c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of
S4.
Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2)
+ AB + BA + B^(2). Hint: Write out 2x2 matrices for A and B using
variable entries in each location and perform the operations on
either side of the equation to determine whether the two sides are
equivalent.
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.
1. Prove that the Cantor set contains no intervals.
2. Prove: If x is an element of the Cantor set, then there is a
sequence Xn of elements from the Cantor set converging
to x.