Overview
This lecture introduces fundamental concepts of sets in mathematics, covering definitions, notation, subsets, cardinality, and well-defined sets through examples and exercises.
Pre-Assessment Review
- A well-defined set has clearly specified elements, e.g., "a set of Philippine presidents."
- The symbol for a null (empty) set is "∅".
- The intersection of sets is indicated by the symbol "∩".
- A subset contains only elements from its parent set.
- The set with cardinality equal to 5 is "the names of the fingers" (thumb, index, middle, ring, pinky).
Definition and Notation of Sets
- A set is a well-defined collection of objects sharing common characteristics.
- Elements are the objects contained in a set.
- Braces { } group the elements of a set; elements are separated by commas.
- Ellipses "..." indicate that a set has many or infinite elements.
- The order of elements does not matter, and each distinct element is listed once.
Well-Defined vs. Not Well-Defined Sets
- Well-defined sets have objective, clear criteria for membership.
- Not well-defined sets are based on subjective qualities (e.g., "famous dancers").
Vocabulary and Symbols in Sets
- Subset: A set whose elements are all contained in another set.
- Universal set: The set under consideration that contains all possible elements.
- Disjoint sets: Two sets with no elements in common.
- Empty set/null set (∅): A set with no elements.
- Cardinal number (cardinality): The number of elements in a set.
Examples and Exercises
- Given set A = {0, 1, 2, ..., 10}, subsets include odd numbers and factors of 10.
- Cardinality of set {a, e, i, o, u} is 5.
- Cardinality of set with repeated letters (e.g., "racer") counts each distinct letter once.
- Finding cardinality examples: "difficulty" (8 distinct letters), odd numbers between 1 and 3 (0 elements), numbers less than 5 (4 elements).
Key Terms & Definitions
- Set — a well-defined collection of objects.
- Element — an object in a set.
- Subset — a set wholly contained within another set.
- Universal Set — the set containing all elements under discussion.
- Empty Set (∅) — a set with no elements.
- Cardinality — the count of elements in a set.
- Disjoint Sets — sets with no shared elements.
- Ellipsis ("...") — notation for continuing elements in a set.
Action Items / Next Steps
- Review set notation and definitions in your module.
- Practice listing elements and finding the cardinality of sets.
- Prepare for the next lesson on set operations and Venn diagrams.