Non-empty vs empty sets – identify a non-empty set: Which of the following sets is non-empty?
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AA = set of odd natural numbers divisible by 2
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BB = {x : x + 5 = 0, x ∈ N}
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CC = set of even prime numbers
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DD = {x : 1 < x < 2, x ∈ N}
Answer
Correct Answer: C = set of even prime numbers
Explanation
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