Set Theory Proofs: Double Inclusion

💡 Core Technique: To prove two sets are equal, show each is a subset of the other!

Notation Clarification: Subset vs Proper Subset

Anderson's Notation (Follow This!)

⊆ means "subset or equal to" (inclusive)

• A ⊆ B means every element of A is in B
• A can equal B

⊂ means "proper subset" (strict)

• A ⊂ B means A ⊆ B AND A ≠ B
• A cannot equal B

Definition: Proper Subset

A ⊂ B if and only if:

1. A ⊆ B (every element of A is in B)
2. AND A ≠ B (A is not equal to B)

Important Note

⚠️ Different mathematicians use different notation!

If you want to use different notation in your work:

• State it clearly at the beginning
• Example: "In this proof, ⊆ denotes subset or equal to"
• Anderson likes when students point out notation inconsistencies politely