Basic Set Operations

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

A set is a collection of distinct objects, called elements.

Basic notation:

x ∈ A : "x is an element of A"
x ∉ A : "x is not an element of A"

Common operations:

Union: A ∪ B = { x : x ∈ A or x ∈ B }
Intersection: A ∩ B = { x : x ∈ A and x ∈ B }
Set difference: A \ B = { x : x ∈ A and x ∉ B }
Complement (relative to a universe U): Aᶜ = U \ A

Two sets A and B are equal (A = B) if they have exactly the same elements:

∀x, x ∈ A ⇔ x ∈ B.

This element-wise perspective is the key to most set proofs.

TL;DR — key idea

Set operations like union, intersection, and complement are defined element-wise: a point belongs to the result exactly when it satisfies the defining condition.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Rewrite the definitions of A ∪ B and A ∩ B in your own words using "and/or" language. Then give an example of two sets A and B and explicitly list A ∪ B and A ∩ B.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.