Set Theory, Probability, Boolean Algebra
∪ vs ∩: What's the Difference?
Clarifying mathematical notation across our complete math symbols knowledge repository. Combining all elements from sets vs. finding strictly shared common elements.
🎯 The Crucial Difference
Union (∪) creates a new set containing every element present in either set (logical OR). Intersection (∩) creates a new set containing only elements present in both sets simultaneously (logical AND).
Detailed Comparison Table
| Feature / Property | ∪ (Set Union (Cup)) | ∩ (Set Intersection (Cap)) |
|---|---|---|
| Logical Equivalent | Logical OR (∨) | Logical AND (∧) |
| Mnemonic Helper | ∪ looks like a 'U' (Union / Bowl collecting all) | ∩ looks like an 'n' (iNtersection / Cap) |
| Size Comparison | |A ∪ B| ≥ max(|A|, |B|) | |A ∩ B| ≤ min(|A|, |B|) |
| Example: {1,2} and {2,3} | {1, 2} ∪ {2, 3} = {1, 2, 3} | {1, 2} ∩ {2, 3} = {2} |
When to Use ∪ (Set Union (Cup))
A ∪ ∅ = AP(A ∪ B) = P(A) + P(B) − P(A ∩ B)Evens ∪ Odds = Integers
When to Use ∩ (Set Intersection (Cap))
A ∩ ∅ = ∅Evens ∩ Odds = ∅ (disjoint){x ∈ ℝ | x > 0} ∩ {x ∈ ℝ | x < 5} = (0, 5)
Common Pitfall to Avoid
Flipping the cup vs cap symbols when working on probability inclusion-exclusion problems.
Help & Insights
∪ vs ∩ Frequently Asked Questions
What is the easy way to remember the difference between ∪ and ∩?
∪ looks like a 'U' (for Union, or a bowl collecting everything from both sets). ∩ looks like an 'n' (for iNtersection, only what is shared in the middle).
What are the Boolean logic equivalents of union and intersection?
Union (∪) corresponds to logical OR (∨) and addition in probability. Intersection (∩) corresponds to logical AND (∧) and joint probability.