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

Study Notes

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.

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).

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

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.