🎯 Logic and Sets Problem Types
📊 Problem Pattern Catalog
Type 1: Basic Set Operations
Pattern: Find union, intersection, or complement of sets Key Concept: Use set operation definitions
Worked Example:
If $A = {1, 2, 3, 4}$ and $B = {3, 4, 5, 6}$, find $A \cup B$ and $A \cap B$.
Solution: $A \cup B = {1, 2, 3, 4, 5, 6}$ (all elements from both sets)
$A \cap B = {3, 4}$ (elements common to both sets)
Type 2: Venn Diagram Problems
Pattern: Use Venn diagrams to solve counting problems Key Concept: Each region represents a specific combination
Worked Example:
In a class of 30 students, 18 like math, 15 like science, and 8 like both. How many like neither?
Solution: Draw a Venn diagram with two circles (Math and Science).
- Math only: $18 - 8 = 10$
- Science only: $15 - 8 = 7$
- Both: $8$
- Neither: $30 - (10 + 7 + 8) = 5$
Type 3: Inclusion-Exclusion Principle
Pattern: Count elements in union of sets Key Formula: $|A \cup B| = |A| + |B| - |A \cap B|$
Worked Example:
A survey of 100 people found that 60 like pizza, 40 like burgers, and 20 like both. How many like at least one?
Solution: $|A \cup B| = |A| + |B| - |A \cap B| = 60 + 40 - 20 = 80$
So 80 people like at least one food.
Type 4: Three-Set Problems
Pattern: Use three-set Venn diagrams and inclusion-exclusion Key Formula: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$
Worked Example:
In a class, 20 students like math, 18 like science, 15 like history, 8 like math and science, 6 like math and history, 5 like science and history, and 3 like all three. How many like at least one subject?
Solution: $|A \cup B \cup C| = 20 + 18 + 15 - 8 - 6 - 5 + 3 = 37$
So 37 students like at least one subject.
Type 5: Logical Statement Analysis
Pattern: Analyze truth values of logical statements Key Concept: Use truth tables or logical reasoning
Worked Example:
If “All birds can fly” is false, what can we conclude about “Some birds cannot fly”?
Solution: If “All birds can fly” is false, then there exists at least one bird that cannot fly.
Therefore, “Some birds cannot fly” is true.
🔍 Problem-Solving Strategy
- Draw a Venn diagram if dealing with sets
- Identify what you know about each set or region
- Use inclusion-exclusion for counting problems
- Check all regions in your diagram
- Verify your answer makes sense
⚠️ Common Mistakes
- Forgetting to subtract intersections in inclusion-exclusion
- Double-counting elements in Venn diagrams
- Missing regions in complex diagrams
- Confusing union and intersection symbols
- Not considering the universal set in complement problems