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
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.
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.
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:
A and B intersecting with no shared elements.
A and B have common elements.
Elements that belong to one set and not the other are shown outside the intersection.
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.
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|.
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.
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|.
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.