Non-empty vs empty sets – identify a non-empty set: Which of the following sets is non-empty?

Introduction / Context:
Non-emptiness requires at least one element satisfies the condition in the specified domain. We check each description and try to produce a witness element or show impossibility.

Given Data / Assumptions:

• N denotes natural numbers
• Odd/even and primality are standard arithmetic properties

Concept / Approach:
(a) Odd numbers cannot be divisible by 2. (b) x = −5 is not natural. (d) No natural number lies strictly between 1 and 2. (c) The even prime 2 exists, so that set is non-empty.

Step-by-Step Solution:
(a) Contradictory property ⇒ empty(b) Solution x = −5 ∉ N ⇒ empty(c) Contains 2 ⇒ non-empty(d) No integer strictly between 1 and 2 ⇒ empty over N

Verification / Alternative check:
List the smallest naturals: 1,2,3. Only 2 is even and prime; it witnesses non-emptiness of (c).

Why Other Options Are Wrong:
They each encode impossible or unsatisfied conditions within the given domains.

Common Pitfalls:
Forgetting that N excludes negatives and that “strict inequalities” remove boundary points.

Final Answer:
C = set of even prime numbers