This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the question about natural numbers: Step 1: Analyze statement I. Statement I says: "The set of natural numbers is closed under addition and multiplication." Natural numbers (usually \1, 2, 3, \ or \0, 1, 2, 3, \) are closed under addition because the sum of any two natural numbers is always a natural number (e.g., 2+3=5). They are also closed under multiplication because the product of any two natural numbers is always a natural number (e.g., 2 × 3=6). Therefore, statement I is correct. Step 2: Analyze statement II. Statement II says: "The set of natural numbers is closed under subtraction." Closure under subtraction means that if you subtract any natural number from another, the result is always a natural number. This is not true. For example, 2 and 5 are natural numbers, but 2-5 = -3, which is not a natural number. Therefore, statement II is incorrect. Step 3: Analyze statement III. Statement III says: "Every non-empty subset of natural numbers has a least element." This is known as the Well-Ordering Principle for natural numbers. Any non-empty collection of natural numbers will always contain a smallest element. For example, in the set \10, 5, 15\, the least element is 5. Therefore, statement III is correct. Step 4: Determine the correct option. Statements I and III are correct, while statement II is incorrect. Comparing this with the given options: (A) Only I (B) I and II (C) I and III (D) II and III The option that includes only I and III is (C). The final answer is C Send me the next one 📸