1c03 textbook

The Language of Sets

Definition of a Set

  • A set is uniquely determined by its elements.

  • The empty set is denoted by ∅ and is called the null set.

  • ∅ is different from {∅};

    • ∅ has no elements, while {∅} is a set containing one element: the empty set itself.

  • Equality of Sets (Definition 4.1.6): Two sets are equal if they have the same elements.

Subsets

  • Definition 4.1.4: Let A and B be sets.

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

    • Symbolically, A ⊆ B if (∀x)[x ∈ A ⇒ x ∈ B].

    • If A is not a subset of B, it is denoted A ⊄ B.

Illustrating Subset Relations

  • Visual representation using circles can help illustrate that A ⊆ B.

Examples of Subsets

  1. Natural Numbers and Integers: N ⊆ Z, Z ⊆ Q, Q ⊆ R.

  2. Sets containing single elements: {{1}, 2} ⊆ {{1}, 2, 3} since {1} and 2 are both included.

  3. A = {n ∈ Z | (∃k ∈ Z) [n = 4k + 1]}, B = {n ∈ Z | n is odd}. Prove A ⊆ B and B ⊆ A.

Axiom of Extensionality

  • This assumption confirms that a set is determined completely by its members.

Conditional Definitions and Examples

  • To illustrate the membership in subsets, computations show member processes, but do not constitute proof.

Formal Proofs of Subsets

  • Goal: Show A ⊆ B using the definitions of A and B, typically through a Given-Goal diagram.

  • An example of a proof with special odd integers shows subsets.

Definitions of Equality in Sets

  • Definition 4.1.6: Let A and B be sets.

    • A = B if (∀x)[x ∈ A ⇔ x ∈ B].

    • A = B iff both A ⊆ B and B ⊆ A exist.

Proper Subsets

  • When A ⊆ B, A may not equal B; it’s expressed as A ⊂ B.

Theorem on Subsets

  • Theorem 4.1.8: If A ⊆ B and B ⊆ C, then A ⊆ C.

The Empty Set

  • The empty set (∅) is contained in every set: ∅ ⊆ A for all sets A.

  • The truth of ∅ ⊆ A as it contains no elements derives that any implication related to it is vacuously true.

Image and Inverse Image

  • Image of a Set: For A ⊆ X and f: X → Y, the image of A is f[A] = {f(x) | x ∈ A}.

  • Preimage of a Set: For B ⊆ Y, the inverse image under f is f −1[B] = {x ∈ X | f(x) ∈ B}.

Functions and Their Sets

  • The mappings defined can relate images of subsets or whole sets; if f maps X to Y, how this makeup can be rationalized using subsets A and B and inverses can assist in graphical or diagrammatic interpretations.

Visualizing Functions and Their Properties

  • Using diagrams and symbols can clarify set operations and function properties (injections, surjections, bijections).

  • Visual proofs support theoretical developments, particularly involving composition and functional outcomes when intersecting or uniting inputs.

Exercises for Practice

  • A variety of exercises facilitate the understanding of set operations, properties of functions, and validating definitions via set examples, challenges, and proofs.