F.Y.B.Sc. MTS 1001 Algebra Notes

Sets

• A set is a collection of distinct objects, referred to as elements or members.
• An empty set is a set with no elements, denoted as ∅.
• A singleton set contains exactly one element.
• Sets can be expressed using the roster method (listing elements) or set-builder form (defining properties of elements).
• A set is nonempty if it has at least one element.
• Subset definition: Set A is a subset of B (A ⊆ B) if every element of A is also in B.
• A superset is the opposite; if A ⊆ B, then B is a superset of A, written as B ⊃ A.
• Two sets are equal if they contain the same elements (A = B).
• A proper subset is denoted A ⊂ B if A is a subset of B and A ≠ B.

Operations on Sets

• The union of sets A and B is defined as A ∪ B = {x : x ∈ A or x ∈ B}.
• Set operations are commutative: A ∪ B = B ∪ A.
• The intersection of sets A and B is A ∩ B = {x : x ∈ A and x ∈ B}.
• Set operations are associative: (A ∪ B) ∪ C = A ∪ (B ∪ C) and similarly for intersection.
• Intersection distributes over union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
• Union distributes over intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
• Two sets are disjoint if their intersection is empty: A ∩ B = ∅.
• The set difference A - B (or A \ B) consists of elements in A not in B.
• The symmetric difference A Δ B combines distinct elements of both sets: A Δ B = (A \ B) ∪ (B \ A).

Universal Set and Complements

• Every set is a subset of some larger set known as the universal set (U).
• The complement of a set A within U is denoted Aᶜ and consists of elements in U that are not in A: Aᶜ = U \ A.
• De Morgan’s Laws for subsets A and B of U:
  • (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
  • (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ

Families of Sets

• A family of sets consists of a collection where each element is itself a set.
• Sets can be indexed by natural numbers, integers, or real numbers, as in intervals I_n = [n, n + 1].
• The union of a family of sets indexed by Γ is defined as ∪α∈Γ F_α = {x ∈ U : ∃α ∈ Γ such that x ∈ F_α}.
• The intersection of a family of sets is defined as ∩α∈Γ F_α = {x ∈ U : ∀α ∈ Γ, x ∈ F_α}.
• A pairwise disjoint family of sets is a collection in which any two distinct sets have no elements in common.