Subsets of finite/infinite sets – find the true statement: Which of the following statements is true?

Introduction / Context:
Basic properties of finiteness and subsets are bedrock facts in set theory and combinatorics. We compare universal quantifier claims to known counterexamples to determine truth values quickly.

Given Data / Assumptions:

• Finite set S has |S| = n for some integer n ≥ 0
• Infinite sets have no finite bound on cardinality

Concept / Approach:
Any subset of a finite set cannot have more elements than S; thus it must also be finite. By contrast, infinite sets can have finite subsets (e.g., {1} ⊂ N), disproving “every subset of an infinite set is infinite.” A proper subset of a finite set cannot be equivalent to the whole (it has strictly fewer elements).

Step-by-Step Solution:
(a) True by definition of finiteness and subset(b) False, counterexample: {1} ⊂ N(c) False, counterexample: N itself is an infinite subset of N(d) False, proper subset has strictly smaller cardinality

Verification / Alternative check:
Cardinality arguments suffice; no computation needed.

Why Other Options Are Wrong:
Each is contradicted by a simple subset counterexample.

Common Pitfalls:
Assuming properties of the whole transfer uniformly to all subsets; infinite sets especially admit both finite and infinite subsets.

Final Answer:
Every subset of a finite set is finite.