Sets – Elements vs subsets inside a mixed set: Given A = {ϕ, {ϕ}, 1, {1, ϕ}, 7}, identify which statement is false.

Introduction / Context:
This question tests the difference between “is an element of” (∈) and “is a subset of” (⊂). Sets that themselves contain sets can cause confusion, so we must check membership literally against the listed elements in A.

Given Data / Assumptions:

• A = {ϕ, {ϕ}, 1, {1, ϕ}, 7}
• Symbols: ϕ denotes the empty set; braces create a set as a single element

Concept / Approach:
Inspect each candidate as an element: A contains ϕ (empty set) and {ϕ} as explicit elements. It contains 1 and {1, ϕ}, but not {1}. For subset checks, a set S is a subset of A if every element of S is in A; since 7 ∈ A and ϕ ∈ A, {7, ϕ} ⊂ A is true.

Step-by-Step Solution:
Check ϕ ∈ A: true (listed)Check {ϕ} ∈ A: true (listed)Check {1} ∈ A: false (A has 1 and {1, ϕ}, not the singleton {1})Check {7, ϕ} ⊂ A: true (both 7 and ϕ are in A)

Verification / Alternative check:
Write A line-by-line and compare literally; braces matter, and {1} is distinct from 1 or {1, ϕ}.

Why Other Options Are Wrong:
Options (a), (b), and (d) are correct statements, so they are not the requested “false” statement.

Common Pitfalls:
Confusing 1 with {1}, or assuming {1} is present because {1, ϕ} is present. Membership is exact, not approximate.

Final Answer:
{1} ∈ A