CF 104053D - Digits
The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning. The error comes from an incorrect application of the quotient definition: it wrongly assumes that $alpha in f/e2$ must itself belong to $f$, which is not required.
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Solve time: 2m 8s
Verified: no
Solution
Correctness
The solution answers all parts, but part (b) is incorrect and breaks subsequent reasoning. The error comes from an incorrect application of the quotient definition: it wrongly assumes that $\alpha \in f/e_2$ must itself belong to $f$, which is not required. This leads to an incorrect result for $f/e_2$, and therefore an incorrect conclusion for $f/(f/e_2)$.
Parts (a), (c), (e) are essentially correct, and (d), (f), (g) are mostly repaired in structure, though (d) still contains informal justification. However, the failure in (b) is decisive.
Gaps and Errors
(b) Incorrect computation of $f/e_2$ (Critical error)
The solution asserts that $\alpha \cup {2} \in f$ must be checked only against elements whose singleton difference $\alpha$ already lies in $f$, and therefore rejects $\alpha = {1}$. This is incorrect.
By definition,
$$\alpha \in f/e_2 \iff \alpha \cap {2} = \varnothing ;\text{and}; \alpha \cup {2} \in f.$$
There is no requirement that $\alpha \in f$.
Checking correctly:
- $\alpha = \varnothing$: ${2} \in f$, valid.
- $\alpha = {1}$: ${1} \cap {2} = \varnothing$ and ${1,2} \in f$, valid.
- $\alpha = {3}, {4}$: fail since unions with ${2}$ are not in $f$.
Thus the correct result is:
$$f/e_2 = {\varnothing, {1}}.$$
The solution’s omission of ${1}$ invalidates all subsequent computations depending on this value.
Summary
The solution is structurally improved, but part (b) contains a fundamental misinterpretation of the quotient definition, producing an incorrect result.
VERDICT: FAIL - incorrect computation of $f/e_2$ due to an invalid extra constraint requiring $\alpha \in f$.