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Unit 1 · Discrete Structures and Optimisation Paper 2 CS · 22 June 2026

Suppose P(·) is used for power set and ∅ is the usual notation of empty set. Arrange the following on the basis of their cardinality in non-decreasing order:
A. P({a, b, {a,b}})
B. P({∅, a, {a}, {{a}}})
C. P(P(∅))
D. P(P({a}))
E. P({{a,b}, c})
Choose the correct answer from the options given below:

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Correct answer: C. C, D, E, A, B

Explanation

Count elements of each inner set first: {a, b, {a,b}} has 3 elements, so |P(A)| = 23 = 8. {∅, a, {a}, {{a}}} has 4 distinct elements, so |P(B)| = 24 = 16. P(∅) = {∅} has 1 element, so P(P(∅)) has 21 = 2 elements. P({a}) = {∅, {a}} has 2 elements, so P(P({a})) has 22 = 4 elements. {{a,b}, c} has 2 elements, so |P(E)| = 22 = 4. So the cardinalities are C=2, D=4, E=4, A=8, B=16, giving the non-decreasing order C, D, E, A, B.

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