Abstract Algebra

1. R, +
   1. Addition on R : ∀x, y ∈ R, x + y ∈ R
   2. Commutative
   3. Associative
2. R, —
   1. Not Commutative
   2. Not Associative
3. R, x
   1. Commutative and Associative
4. R, /
   1. Division on R is NOT a binary operation because ∀x ∈ R ∃ 0 ∈ R s.t. x/0 is undefined (not an element of R)

Anlagen:

• Set Operations

1. (x + y) + z = x + (y + z)

1. Let A be a set and let P(A) be its power set. (P(A), ∩) and (P(A), ∪) are both semigroups.
2. (Mn, ∗), the set of n*n matrices with entries in R with the operation of matrix multiplication (which is associative) forms a semigroup.
3. Let (A, ∗) be a semigroup

1. Let (A, ∗) be a semigroup. ∀a1, ..., an ∈ A, a1 ∗ a2 ∗ ... ∗ an has the same value regardless of how the product is bracketed.

1. An element, e, e ∈ A, is called an identity element for the binary operation ∗, if e ∗ x = x ∗ e = x, ∀x ∈ A.
   1. (R, +): e = 0
   2. (R, x): e = 1
   3. (P(A), ∪): e = ∅
   4. (P(A), ∩): e = A
   5. (Mn, ∗) has In, the identity matrix, as its identity element

1. (R, +): commutative monoid, e = 0
2. (R, x): commutative monoid, e = 1
3. (P(A), ∪): commutative monoid, e = ∅
4. (N, +): commutative monoid, e = 0

1. (Mn, ∗): monoid, e = In, but matrix multiplication is not commutative
2. (N*, +): semigroup, as N* = N\{0}

1. (R, +): invertible, e = 0, ∀x ∈ R,(−x) is the inverse of xas x + (-x) = (-x) + x = 0.

1. (R, x): invertible, e = 1, ONLY if x != 0. If x != 0, inverse of x is 1/x, since x * 1/x = 1/x * x = 1
2. (Mn, ∗): invertible, e = In, ONLY if det(A) != 0

1. (P(A), ∪): non-invertible, e = ∅
2. However, the element ∅ of P(A) is invertible and has itself as its inverse: ∅ ∪ ∅ = ∅ ∪ ∅ = ∅

1. Closure
   1. IMPLICIT as ∀a, b ∈ A, a ∗ b ∈ A
2. Commutative
   1. (R, +, 0): -x (inverse)
   2. (Q*, x, 1), 1/q (inverse)
   3. (R^3, +, 0) vectors, -x, -y, -z inverse
      1. (x, y, z) + (x', y', z') = (x + x', y + y', z + z')

1. f(x∗y) = f(x) ∗ f(y) ∀x, y ∈ A