Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b}
a. Prove directly that H is a normal subgroup in G (use the fact
that closed under composition and inverses)
b. Prove that G/H is isomorphic to Z8.
c. What is the index of [G : H]?
Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈
Z,b ∈ N} (recall that [(a,b)] = {(c,d) ∈Z×N : (a,b) ∼ (c,d)}).
We define an addition and a multiplication on X as follows:
[(a,b)] + [(c,d)] = [(ad + bc,bd)] and [(a,b)]·[(c,d)] =
[(ac,bd)]
Prove that this addition and multiplication is well-defined on
X.
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
Let A be an m x n matrix and b and x be vectors such that
Ab=x.
a) What vector space is x in?
b) What vector space is b in?
c) Show that x is a linear combination of the columns of A.
d) Let x' be a linear combination of the columns of A. Show that
there is a vector b' so that Ab' = x'.
Suppose A = {(a, b)| a, b ∈ Z} = Z × Z. Let R be the relation
define on A where (a, b)R(c, d) means that 2 a + d = b + 2 c.
a. Prove that R is an equivalence relation.
b. Find the equivalence classes [(−1, 1)] and [(−4, −2)].
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)
Let G be a group. For each x ∈ G and a,b ∈ Z+
a) prove that xa+b = xaxb
b) prove that (xa)-1 = x-a
c) establish part a) for arbitrary integers a and b in Z
(positive, negative or zero)