Translate ∀x∃y(x<y) in English. Consider domain as a real number for both the…
2024
Translate ∀x∃y(x<y) in English. Consider domain as a real number for both the variables.
- A.
For all real numbers x, there exists a real number y such that x is less than y
- B.
For every real numbers y, there exists a real number x such that x is less than y
- C.
For some real numbers x, there exists a real number y such that x is less than y
- D.
For each and every real numbers x and y, x is less than y
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Correct answer: A
Translation: For every real number x, there exists a real number y such that x < y.
Reasoning (constructive proof):
Let x be an arbitrary real number.
Choose y = x + 1.
Then y - x = 1 > 0, so x < y. Since x was arbitrary, for every real x there exists such a y.
Conclusion: The formula is correctly translated as above and is true over the real numbers by the explicit choice y = x + 1.
Note: Changing the order of quantifiers or replacing a universal quantifier with an existential one produces a different statement; such changes are not equivalent to the original formula.
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