Recording-2025-02-03T21:45:02.802Z

Sets and Set Notation

  • Definition of a set: A collection of distinct objects or elements.

  • Set membership: An element is in a set if it satisfies a property.

  • Notation:

    • Element in set: a ∈ A

    • Element not in set: a ∉ A

Propositional Functions

  • A propositional function is a function that returns truth values based on the value of variables.

    • Example: P(n): n > 10 (where P is a propositional function).

    • Specific case: P(1) = False, as 1 is not greater than 10.

  • Formulating propositional functions:

    • Example: Let A = {n ∈ Z | n is even and n > 10}

    • This states that A contains integers that are even and greater than 10.

Set Relationships

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

    • Example: If A = {1, 2} and B = {1, 2, 3}, then A is a subset of B.

  • Strict subset: A is a strict subset of B if A is a subset of B and A ≠ B.

  • Intersection of sets: Elements common to both sets.

Visualizing Sets

  • Diagrams demonstrate hierarchy and relationships between sets:

    • A subset is visually represented as a smaller circle within a larger circle (set).

    • Elements outside the subset represent elements in the superset but not in the subset.

  • Combinations of sets may be expressed visually to clarify relationships:

  1. A and B intersecting with no shared elements.

  2. A and B have common elements.

  3. Elements that belong to one set and not the other are shown outside the intersection.

Equality of Sets

  • Two sets A and B are equal (A = B) if:

    • A is a subset of B and B is a subset of A.

    • Notation: A ≡ B indicates equal sets.

Infinite Sets

  • Countable vs. uncountable sets.

    • Countable: Natural numbers (N), can be enumerated or paired with integers.

    • Uncountable: Sets that cannot be enumerated (like the set of real numbers).

  • Cardinality: A measure of the number of elements in a set, denoted by |A|.

Power Sets

  • The power set of a set A is the set of all possible subsets of A.

    • Notation: P(A) = Set of all subsets of A.

    • If A contains n elements, the power set contains 2^n elements.

Cartesian Products

  • The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is from A and b is from B.

  • Size of the Cartesian product: |A x B| = |A| * |B|.

Key Takeaways

  • Understand the symbols and terminology associated with set theory (subset, intersection, power set).

  • Practice using propositional functions to define sets and understand how to visualize them.

  • Apply concepts of countable and uncountable sets in relation to programming data structures.

robot