Empty Set
or {}.Worked Example
Problem: Let A = {1, 2, 3} and B = {4, 5, 6}. Find the intersection A ∩ B.
Step 1: The intersection of two sets contains only the elements that appear in both sets.
A∩B={x:x∈A and x∈B}
Step 2: Check each element of A to see if it also belongs to B. The number 1 is not in B, 2 is not in B, and 3 is not in B. No element is shared.
Step 3: Since no elements are common to both sets, the intersection is the empty set.
A∩B=∅
Answer: A ∩ B = ∅. The two sets share no elements, so their intersection is the empty set.
Why It Matters
The empty set serves as the foundation of set theory, much like zero does in arithmetic. It is a subset of every set, which makes many proofs and definitions in mathematics cleaner and more consistent. Whenever you solve a system of equations that has no solution, or find no values satisfying an inequality, the solution set is the empty set.
Common Mistakes
Mistake: Writing {∅} instead of ∅ for the empty set.
Correction: The notation {∅} is a set containing one element (the empty set itself), so it is not empty. Use ∅ or {} to denote the set with no elements.
Related Terms
- Set — The general concept that the empty set belongs to
- Element of a Set — The empty set has no elements
- Subset — The empty set is a subset of every set
- Intersection — Disjoint sets have an empty intersection
- Null Set — Another name for the empty set