Abstract algebra 1_compressed

Page 1: Sets and Equivalence Relations

• Definition of a Set

  • A set S consists of elements; if a is an element of S, it's denoted by a ∈ S.

• Empty Set

  • There exists one set with no elements, the empty set, denoted by ∅ or {}.

• Well-Defined Sets

  • A set S is well-defined if for any object a, it is clear if a is in S (a ∈ S) or not (a ∉ S).

• Subsets

  • A set B is a subset of set A (B ⊆ A) if every element of B is also in A.

  • Notations: B ⊆ A or A ⊇ B.

  • Note: For any set A, A itself and ∅ are subsets of A.

Page 2: Definitions and Sets

• Improper Subset

  • A set A is an improper subset of itself; all other subsets are proper.

• Well-Defined Collection of Sets

  • Examples of sets include:

    • Complex numbers (ℂ)

    • Real numbers (ℝ)

    • Rational numbers (ℚ)

    • Integer numbers (ℤ)

    • Natural numbers (ℕ)

    • Irrational numbers (ℝ \ ℚ)

    • The sets of all positive integers and non-zero real numbers.

Page 5: Binary Operations

• Definition

  • A binary operation on a set S is a function from S × S to S.

• Notation

  • Denote binary operation as (a, b)

    • Examples include addition (+) and multiplication (×).

• Checking Binary Operations

  1. For ℝ:

     • (a, b) → a + b, which is binary.

  2. For ℝ:

     • (a, b) → a × b, which is binary.

  3. For ℤ:

     • (a, b) → a − b is not binary since it can yield elements outside of ℤ.

Page 7: Properties of Binary Operations

• Commutativity

  • A binary operation is commutative if a × b = b × a for all a, b.

• Associativity

  • An operation is associative if (a × b) × c = a × (b × c) for all a, b, c.

Page 10: Concepts of Groups

• Definition of a Group

  • A group (G, x) consists of a set G and a binary operation x that meets certain axioms:

    1. Binary operation is associative.

    2. There is an identity element in G.

    3. Each element in G has an inverse under the operation.

• Example

  • Let * defined under Q+ as a*b = ab.

  • Check properties:

    • Closed: If a, b ∈ Q+, ab ∈ Q+.

    • Associative: Demonstrated by a, b, c ∈ Q+.

Page 12: Identity and Inverse Elements

• Identity Element

  • If S is a set with a binary operation, e is an identity if for all x ∈ S, ex = x and xe = x.

• Inverse Element

  • An element a' is an inverse of a if a * a' = e.

Page 18: Order of Groups

• Definition

  • The order of a group G is defined as the number of elements in G, denoted |G|.

  • For finite groups, the possible orders of subgroups relate to factors of |G|.

Page 22: Isomorphic Groups

• Definition

  • Two groups G and H are isomorphic if there is a bijective homomorphism between them.

Page 64: Lagrange's Theorem

• Theorem

  • If H is a subgroup of finite group G, then the order of H divides the order of G.

Page 84: Kernel of a Homomorphism

• Definition

  • The kernel of a homomorphism d: G → H consists of elements x ∈ G such that d(x) = e_H (the identity in H).

• Significance

  • Indicates a normal subgroup if the kernel is a subgroup of G.