Lesson subsection
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.
Best flow: read → think on paper → write a short explanation → refine with feedback.
To prove two sets A and B are equal, the standard method is:
Together, these give A = B.
To show A ⊆ B:
Example:
Prove: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
You would prove:
Most set equalities are really just structured logical equivalences.
TL;DR — key idea
To prove A = B, show A ⊆ B and B ⊆ A by starting with an arbitrary element and chasing membership through definitions.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Outline a proof (you don’t have to fill every detail) that: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Write the two directions: 1. Assume x ∈ A ∪ (B ∩ C) and show x ∈ (A ∪ B) ∩ (A ∪ C). 2. Assume x ∈ (A ∪ B) ∩ (A ∪ C) and show x ∈ A ∪ (B ∩ C).
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
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.