Power set of a singleton: Write the power set of {0}.

Introduction / Context:
The power set P(S) is the set of all subsets of S. For a singleton set {x}, its power set always contains exactly two subsets: the empty set and the singleton itself. This is a foundational fact that generalizes to |P(S)| = 2^n for any finite set S with n elements.

Given Data / Assumptions:

• S = {0}
• Empty set ϕ is a subset of every set

Concept / Approach:
List all subsets of {0}. Since there is only one element, the subsets are: ϕ (take none) and {0} (take the single element). Collect these subsets into a set to obtain the power set.

Step-by-Step Solution:
Possible selections of elements from {0}: none or {0}Hence P({0}) = {ϕ, {0}}

Verification / Alternative check:
Count check: |{0}| = 1, so |P({0})| = 2^1 = 2 subsets, matching {ϕ, {0}}.

Why Other Options Are Wrong:
‘ϕ’ alone and ‘{0}’ alone list single subsets, not the set of all subsets; ‘{ϕ}’ misses {0}.

Common Pitfalls:
Confusing an element with a subset or listing elements instead of collecting them into the power set. Remember, the power set is itself a set whose elements are subsets.

Final Answer:
{ϕ, {0}}