🧠 Logic and Sets Reference
🎯 Key Concepts
Set
A collection of distinct objects, called elements.
Element
An individual object that belongs to a set.
Subset
A set where every element is also in another set.
Union
The set of all elements that are in either set A or set B.
Intersection
The set of all elements that are in both set A and set B.
📐 Set Notation
Set Builder Notation
$${x : \text{condition}}$$
Usage: Define a set by describing its elements Example: ${x : x \text{ is an even number}}$ = set of even numbers
Roster Notation
$${a, b, c, d}$$
Usage: List all elements of a set Example: ${2, 4, 6, 8}$ = set of even numbers from 2 to 8
Empty Set
$$\emptyset \text{ or } {}$$
Usage: Set with no elements Example: Set of odd numbers that are also even = $\emptyset$
🔗 Set Operations
Union
$$A \cup B = {x : x \in A \text{ or } x \in B}$$
Usage: Combine elements from both sets Example: ${1, 2, 3} \cup {3, 4, 5} = {1, 2, 3, 4, 5}$
Intersection
$$A \cap B = {x : x \in A \text{ and } x \in B}$$
Usage: Find elements common to both sets Example: ${1, 2, 3} \cap {3, 4, 5} = {3}$
Complement
$$A’ = {x : x \notin A}$$
Usage: Find elements not in set A Example: If universal set is ${1, 2, 3, 4, 5}$ and $A = {1, 3}$, then $A’ = {2, 4, 5}$
🎨 Venn Diagrams
Basic Venn Diagram
- Two circles for two sets
- Overlapping region for intersection
- Non-overlapping regions for elements in only one set
Three-Set Venn Diagram
- Three circles for three sets
- Seven regions total
- Center region for intersection of all three sets
🧮 Counting with Sets
Inclusion-Exclusion Principle
$$|A \cup B| = |A| + |B| - |A \cap B|$$
Usage: Count elements in union of two sets Example: If $|A| = 10$, $|B| = 15$, and $|A \cap B| = 3$, then $|A \cup B| = 10 + 15 - 3 = 22$
Three-Set Inclusion-Exclusion
$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$
Usage: Count elements in union of three sets Note: Add all sets, subtract all pairwise intersections, add back triple intersection
💡 Problem-Solving Strategies
- Draw a Venn diagram to visualize the problem
- Identify what you know about each set
- Use inclusion-exclusion for counting
- Check your work by verifying the diagram
- Consider all regions in the diagram
⚠️ Common Mistakes
- Forgetting to subtract intersections in inclusion-exclusion
- Double-counting elements in Venn diagrams
- Missing regions in complex Venn diagrams
- Confusing union and intersection symbols
- Not considering the universal set in complement problems