Sets: Definition, Cardinality, Subsets

• A set is a well-defined collection of objects, which are called elements or members.
• Sets are denoted by capital letters and described by listing elements in curly braces.
• If x is an element of set A, it's written as x ∈ A; if y is not, it's written as y ∉ A.
• Two sets A and B are equal (A = B) if they contain the same elements, regardless of order or repetition.
• The set of all rational numbers (Q) is written as {a/b | a, b are integers, b ≠ 0}.

Cardinality

• Cardinality is the size of a set denoted by |A|.
• The empty set (Ø) has a cardinality of 0.
• Sets can have an infinite number of elements (integers, prime numbers, odd numbers).

Subsets and Proper Subsets

• If every element of A is also in B, A is a subset of B, written A ⊆ B.
• B is a superset of A, written B ⊇ A.
• A proper subset (A ⊂ B) means A is strictly contained in B, excluding at least one element.
• The empty set is a subset of every set.
• Every set is a subset of itself

Intersections and Unions

• The intersection of A and B (A ∩ B) contains elements in both A and B.
• Disjoint sets have an intersection of 0.
• The union of A and B (A ∪ B) contains elements in A or B or both.
• A ∪ B = B ∪ A (commutative property of union).
• A ∩ B = B ∩ A (commutative property of intersection).
• A ∪ 0 = A (identity property of union).
• A ∩ 0 = 0 (identity property of intersection).

Complements

• The relative complement of A in B (B \ A) is the set of elements in B but not in A : B \ A = {x ∈ B | x ∉ A}
• A \ A = 0
• A \ 0 = A
• 0 \ A = 0

Significant Sets

• N: natural numbers {0, 1, 2, 3,...}.
• Z: integers {..., -2, -1, 0, 1, 2,...}.
• Q: rational numbers {a/b | a, b ∈ Z, b ≠ 0}.
• R: real numbers.
• C: complex numbers.

Products and Power Sets

• The Cartesian product of A and B (A × B) forms pairs with the first element from A and the second from B : A × B = {(a,b) | a ∈ A, b ∈ B}
• The power set of S (P(S)) is the set of all subsets of S : {T | T ⊆ S}
• If |S| = k, then |P(S)| = 2ᵏ.

Sums and Products

• Sums of multiple terms can be written concisely : ∑ᵢ₌₁ⁿ i
• ∑ᵢ₌ₘⁿ f(i) represents the sum f(m) + f(m+1) + ... + f(n)
• Products of multiple terms can be written concisely : Πᵢ₌₁ⁿ i
• Πᵢ₌ₘⁿ f(i) represents the product f(m)f(m+1)...f(n)

Universal and Existential Quantifiers

• The universal quantifier (∀) means "for all."
• (∀n ∈ N)(n² + n + 41 is prime) means for all natural numbers n, n² + n + 41 is prime.
• The existential quantifier (∃) means "there exists."
• (∃x ∈ Z)(x < 2 and x² = 4) means there is an integer x less than 2, where x² equals 4.
• (∀x ∈ Z)(∃y ∈ Z)(y > x) means for every integer, a larger integer can be found.
• (∃y ∈ Z)(∀x ∈ Z)(y > x) means there is a largest integer.