Session 3A
Set notation and subset proofs
45-75 min - work through lesson notes, practice, and the MCQ check.
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Session tips
- Finish the MCQ before marking complete.
- Check the answer outlines after trying on paper.
- Mark complete to update your certificate progress.
Lesson notes
- Set proofs are usually done by element-chasing, which means you prove membership statements by tracking what it means to belong to a set. To prove A subseteq B, you start with "Let x be an arbitrary element of A" and then use the definition of A to show x must be in B. This is the set version of a direct proof: assume membership in the left set, derive membership in the right set. Similarly, to prove two sets are equal, you show both inclusions: A subseteq B and B subseteq A.
- The operations union, intersection, and difference translate into logical connectors. x in A union B means x in A or x in B. x in A intersect B means x in A and x in B. x in A \ B means x in A and not in B. Once you write membership in logic form, the proof often becomes a logic exercise. The main discipline is writing each step explicitly: state what membership gives you, transform it, then repackage it as membership in the target set.
Worked examples
- A intersect B subseteq A: if x in A intersect B, then x in A.
Practice
- 1Prove: A intersect B subseteq B
- 2Prove: A subseteq A union B
- 3Prove: If A subseteq B and B subseteq C, then A subseteq C
MCQ
To prove A subseteq B, you should start with: