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5. The following statement is false, write the negation and disprove it using a counter example: VxeR, if x< 2 then x² <

5. The following statement is false, write the negation and disprove it using a counter example: VxeR, if x< 2 then x² <

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 5:** The following statement is false; write the negation and disprove it using a counterexample: ∀ x ∈ ℝ, if x < 2 then x² < 4 **Explanation:** This statement claims that for all real numbers x, if x is less than 2, then the square of x is less than 4. To disprove it, we must find a counterexample where a real number x is less than 2, but x² is not less than 4. **Negation:** The negation of the statement is: There exists some x ∈ ℝ such that x < 2 and x² ≥ 4. **Counterexample:** Consider x = -3. While it is true that -3 < 2, we find that (-3)² = 9, which is greater than 4. Thus, this serves as a counterexample to the original statement.
Transcribed Image Text:**Problem 5:** The following statement is false; write the negation and disprove it using a counterexample: ∀ x ∈ ℝ, if x < 2 then x² < 4 **Explanation:** This statement claims that for all real numbers x, if x is less than 2, then the square of x is less than 4. To disprove it, we must find a counterexample where a real number x is less than 2, but x² is not less than 4. **Negation:** The negation of the statement is: There exists some x ∈ ℝ such that x < 2 and x² ≥ 4. **Counterexample:** Consider x = -3. While it is true that -3 < 2, we find that (-3)² = 9, which is greater than 4. Thus, this serves as a counterexample to the original statement.
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