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`.

The least you need to know

Set equality means the sets contain exactly the same elements.
A standard proof is to show `A ⊆ B` and `B ⊆ 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

Tiny worked example

To show `A∩B = B∩A`, take any `x ∈ A∩B`.
Then `x ∈ A` and `x ∈ B`, so `x ∈ 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 this kind of problem

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.