Identify the empty set among described sets: Which of the following sets is empty (has no elements)?

Introduction / Context:
Empty sets arise when stated conditions have no solution in the specified domain. We evaluate each description carefully against its domain—naturals or integers—and ordinary real-world sets like month names.

Given Data / Assumptions:

• N denotes natural numbers
• Integers include negative numbers, zero, and positives
• Calendar months are standard English names

Concept / Approach:
Test each option: (a) has at least 1; (b) equation 3x + 1 = 0 gives x = −1/3, not natural; (c) integers −1 < x < 1 gives x = 0; (d) “February” begins with F, so at least one month qualifies.

Step-by-Step Solution:
(a) x ≤ 1 in N ⇒ {1} at least ⇒ non-empty(b) 3x + 1 = 0 ⇒ x = −1/3 ∉ N ⇒ empty(c) Integers with −1 < x < 1 ⇒ {0} ⇒ non-empty(d) Months beginning with F ⇒ {February} ⇒ non-empty

Verification / Alternative check:
Domains matter: the same equation in reals would not be empty, but in naturals it is.

Why Other Options Are Wrong:
They each have at least one element: 1, 0, or “February”.

Common Pitfalls:
Forgetting domain restrictions (N vs Z vs R) and assuming solutions exist when they do not.

Final Answer:
B = {x : 3x + 1 = 0, x ∈ N}