g10-01-02 · Core topics
Containment, equality and the empty set
When is a subset proper?
1 MIN · PREDICT ON PAPER
Compare A={1,…,a} and B={1,…,b}; predict the relation when a=b or a=0.
Discuss or write first. No typed answer is required to continue.
A⊆B: true; A⊊B: true.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
A⊆B allows equality. A proper subset additionally requires an element of B outside A. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: a and b are integers from 0 to 8; an upper bound of zero denotes the empty set.
Mathematical justification: For these initial segments, a≤b gives inclusion; a>b leaves a in A outside B. Equality holds iff a=b.
1 MIN · EXIT CHECK
Is ∅ a proper subset of {1}? Is ∅ a proper subset of ∅?
Check after answering
Yes; no. The second pair is equal.
Teacher guidance and extension
Before: Set notation
Ask: Request an explicit witness whenever inclusion fails.
Watch for: An empty set is a subset of itself but not a proper subset of itself.
Challenge: Describe all pairs (a,b) giving A⊊B without listing every set.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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