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.