Decide whether or not the following propositions are true or false. Justify each of your conclusions with a proof or a counterexample. (d) ∃x ∈ R ¬(x 2 < 36) (e) ¬∀x ∈ R (x 2 < 36) (f) ∀x ∈ R ¬(x 2 < 36)

Decide whether or not the following propositions are true or false. Justify each of your conclusions with a proof or a counterexample. (d) ∃x ∈ R ¬(x 2 < 36) (e) ¬∀x ∈ R (x 2 < 36) (f) ∀x ∈ R ¬(x 2 < 36)

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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Question

Decide whether or not the following propositions are true or false. Justify each of
your conclusions with a proof or a counterexample.

(d) ∃x ∈ R ¬(x 2 < 36)
(e) ¬∀x ∈ R (x 2 < 36)
(f) ∀x ∈ R ¬(x 2 < 36)

Expert Solution

Step 1

(d) ∃x ∈ R ¬(x² < 36)

≈ ∃x ∈ R (x² ≥ 36)

Consider any x ≥6 or x ≤ -6

Then, x²≥ 36 => ¬(x² < 36) becomes true.

For example, for x = 7, x²= 49 > 36

Hence, the given proposition is True. [Ans]

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