Overview
This lecture explains the difference between inclusive "or" and exclusive "or" in word problems and how to solve problems involving each type.
Types of "Or" in Word Problems
- "Or" has two meanings: inclusive (A or B or both) and exclusive (A or B, but not both).
- Restaurant example: "soup or salad" = exclusive or; "coffee or dessert" = inclusive or.
- In math word problems, assume "or" means inclusive unless clearly stated otherwise.
Inclusive Or (Union)
- Inclusive or means select elements that are in A, B, or both (A ∪ B).
- Example: To find countries with flags containing red, white, or blue, use the union of sets R, W, and B (R ∪ W ∪ B).
Exclusive Or (Either, But Not Both)
- Exclusive or applies when the problem says "one or the other, but not both".
- Exclusive or includes elements in A but not B (A \ B), and in B but not A (B \ A).
- Mathematically, exclusive or = (A \ B) ∪ (B \ A) or (A ∪ B) \ (A ∩ B).
Example: Multiples of Three or Five
- Task: Find numbers between 10 and 30 that are multiples of 3 or 5, but not both (exclusive or).
- A = Multiples of 3: 12, 15, 18, 21, 24, 27, 30
- B = Multiples of 5: 10, 15, 20, 25, 30
- A \ B = 12, 18, 21, 24, 27
- B \ A = 10, 20, 25
- Final answer: 10, 12, 18, 20, 21, 24, 25, 27
- Alternatively: (A ∪ B) \ (A ∩ B) = remove common elements (15, 30) from the union.
Key Terms & Definitions
- Inclusive Or (Union) — Means "A or B or both" (A ∪ B).
- Exclusive Or — Means "A or B, but not both" ((A \ B) ∪ (B \ A) or (A ∪ B) \ (A ∩ B)).
- Set Difference (A \ B) — Elements in A that are not in B.
- Intersection (A ∩ B) — Elements common to both A and B.
Action Items / Next Steps
- Review how to compute unions, intersections, and set differences.
- Watch the next video to learn about interpreting "but not" in word problems.