What are Union and Intersection?
Union and intersection are fundamental operations in set theory that allow us to combine or find common elements between sets. These operations are essential when working with interval notation, solving compound inequalities, and analyzing mathematical relationships.
Union (∪)
The union of two sets A and B, written as A ∪ B, contains all elements that are in A, in B, or in both. It represents the "OR" operation.
Think of union as combining all elements from both sets.
Intersection (∩)
The intersection of two sets A and B, written as A ∩ B, contains only the elements that are in both A and B. It represents the "AND" operation.
Think of intersection as finding common elements between sets.
Union and Intersection Symbols
Union Symbol: ∪
- • Pronounced as "union"
- • Represents "OR"
- • Combines all elements
- • Example: A ∪ B
Intersection Symbol: ∩
- • Pronounced as "intersection"
- • Represents "AND"
- • Finds common elements
- • Example: A ∩ B
Union and Intersection with Interval Notation
When working with intervals, union and intersection operations help us combine or find overlaps between ranges. This is particularly useful when solving compound inequalities.
Union of Intervals
Example 1: Disjoint Intervals
Find the union of (-∞, -2) and (3, ∞):
Solution: (-∞, -2) ∪ (3, ∞)
This represents all numbers less than -2 OR greater than 3.
In words: All real numbers except those between -2 and 3 (inclusive).
Example 2: Overlapping Intervals
Find the union of [1, 5] and [3, 8]:
Solution: [1, 5] ∪ [3, 8] = [1, 8]
When intervals overlap, the union is the interval from the smallest start to the largest end.
Intersection of Intervals
Example 3: Overlapping Intervals
Find the intersection of [1, 5] and [3, 8]:
Solution: [1, 5] ∩ [3, 8] = [3, 5]
The intersection is the overlap: numbers that are in both intervals.
Example 4: Disjoint Intervals
Find the intersection of (-∞, -2) and (3, ∞):
Solution: (-∞, -2) ∩ (3, ∞) = ∅ (empty set)
When intervals don't overlap, their intersection is empty.
Example 5: Nested Intervals
Find the intersection of [2, 10] and [4, 6]:
Solution: [2, 10] ∩ [4, 6] = [4, 6]
When one interval is completely inside another, the intersection is the smaller interval.
Visual Representation: Venn Diagrams
Venn diagrams provide a visual way to understand union and intersection:
Union Visualization
Union includes all shaded areas (both circles)
Intersection Visualization
Intersection is only the overlapping area (dark center)
Properties of Union and Intersection
| Property | Union | Intersection |
|---|---|---|
| Commutative | A ∪ B = B ∪ A | A ∩ B = B ∩ A |
| Associative | (A ∪ B) ∪ C = A ∪ (B ∪ C) | (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| Distributive | A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) | A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| Identity | A ∪ ∅ = A | A ∩ U = A |
Union and Intersection in Compound Inequalities
Union and intersection directly correspond to OR and AND in compound inequalities. These operations are also essential when expressing domain and range of functions:
OR Inequalities → Union
The inequality x < -1 OR x > 4 corresponds to:
(-∞, -1) ∪ (4, ∞)
Union combines both solution sets.
AND Inequalities → Intersection
The inequality -3 < x AND x < 5 corresponds to:
(-∞, 5) ∩ (-3, ∞) = (-3, 5)
Intersection finds the overlap (common solution).
Step-by-Step: Finding Union and Intersection
For Union (∪):
- List all intervals involved
- If intervals overlap, combine them into a single interval
- If intervals are disjoint, keep them separate with ∪
- Write the result in interval notation
For Intersection (∩):
- Identify the overlapping region
- Find the maximum of the left endpoints
- Find the minimum of the right endpoints
- If there's overlap, that's your intersection
- If no overlap, intersection is empty (∅)
Practice Problems
Problem 1
Find the union: [1, 4] ∪ [3, 6]
Show Answer
Answer: [1, 6]
The intervals overlap, so combine from 1 to 6.
Problem 2
Find the intersection: (-2, 5) ∩ [0, 8)
Show Answer
Answer: [0, 5)
The overlap is from 0 (included) to 5 (excluded).
Problem 3
Find the union: (-∞, -1) ∪ (2, 5] ∪ [7, ∞)
Show Answer
Answer: (-∞, -1) ∪ (2, 5] ∪ [7, ∞)
All intervals are disjoint, so keep them separate.
Common Applications
Solving Inequalities
Union and intersection help express solutions to compound inequalities clearly.
Function Analysis
Finding where functions are defined or have specific properties often involves union/intersection.
Probability
Union and intersection are fundamental in probability theory for combining events.
Database Queries
SQL and database operations use union and intersection concepts for combining data sets.
Tips for Working with Union and Intersection
- ✓Visualize first: Draw number lines or Venn diagrams to understand the relationship
- ✓Check for overlap: Always determine if intervals overlap before combining
- ✓Use interval notation: Express results clearly using interval notation
- ✓Remember the symbols: ∪ means "or" (union), ∩ means "and" (intersection)
- ✓Simplify when possible: Combine overlapping intervals in unions
Conclusion
Understanding union and intersection is essential for working with sets, solving compound inequalities, and expressing solutions using interval notation. These operations appear throughout mathematics, from basic algebra to advanced set theory.
Remember that union (∪) combines sets (OR), while intersection (∩) finds common elements (AND). Practice with different intervals, visualize with number lines, and you'll master these fundamental operations.