Proof / Set Equality
Least You Need to Know: Proving Set Equality
To prove two sets are equal, show both inclusions or use a membership argument with x ∈ A iff x ∈ B.
Least you need to know
- Set equality means the sets contain exactly the same elements.
- A standard proof is to show
A ⊆ BandB ⊆ A. - Element-chasing works by starting with an arbitrary element.
- One example is never enough to prove two sets are equal.
Key notation
- A ⊆ B — every element of A is in B
- A = B — A ⊆ B and B ⊆ A
- x ∈ A — x is an element of A
Worked example
- To show
A∩B = B∩A, take anyx ∈ A∩B. - Then
x ∈ Aandx ∈ B, sox ∈ B∩A. - Reverse the argument for the other inclusion.
Common mistakes
- Students often prove only one inclusion.
- Students often talk about examples instead of arbitrary elements.
- Students often confuse union and intersection conditions.
How to recognize it
- If a set identity looks symmetric, element-chasing is usually clean.
- Write what membership in the left side means before rewriting it.
- For equality, ask what still needs to be shown after the first inclusion.
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