Overview
This lecture covers Venn diagrams, including how to fill them with numbers, interpret set notation (union, intersection, complement), and solve probability questions using these diagrams.
Venn Diagram Basics
- A Venn diagram visually represents sets and their relationships within a universal set.
- The universal set (∪) contains all elements under consideration (e.g., all odd numbers less than 30).
- Each number appears only once in the Venn diagram.
Filling Venn Diagrams
- List all elements of the universal set before placing them in the circles.
- Placing numbers: start with intersections (numbers in multiple sets), then fill individual sets, and finally, numbers outside all sets.
- For three-set diagrams, handle intersections methodically: all three sets, then pairs, then singles.
Set Notation and Symbols
- Union (A ∪ B): Elements in A or B (or both); interpreted as "A or B".
- Intersection (A ∩ B): Elements in both A and B; interpreted as "A and B".
- Complement (A'): Elements not in A.
Probability with Venn Diagrams
- Probability = (number of favorable outcomes) / (total outcomes in universal set).
- For A ∪ B: Count all numbers in either circle.
- For A ∩ B: Count only numbers in the intersection.
- For A': Count numbers not in circle A, including those outside both circles.
Example Problems
- For sets A and B from odd numbers less than 30:
- A ∪ B: 7/15 (all numbers within either circle).
- A ∩ B: 2/15 (numbers in both circles).
- A': 9/15 (numbers not in A).
- For three-set diagrams with even numbers between 1 and 25:
- A ∩ B: 2/12 (numbers in both A and B, including those in all three).
- B ∪ C: 8/12 (any number in B, C, or both).
- C': 7/12 (numbers not in C).
Key Terms & Definitions
- Universal Set — All possible elements considered in the problem.
- Union (A ∪ B) — All elements in A or B or both.
- Intersection (A ∩ B) — Elements in both A and B.
- Complement (A') — Elements not in set A.
- Probability — The chance of an event, calculated as favorable outcomes over total outcomes.
Action Items / Next Steps
- Practice filling Venn diagrams and calculating probabilities for provided exercises.
- Review set notation and ensure you can identify unions, intersections, and complements.
- Prepare for next lesson on more complex Venn diagram problems involving algebraic reasoning.