Use truth sets to prove or disprove the following quantified statements.Note: You can view the solutions by clicking on "Show Detailed Solution" at the end of the problem.d) Let D = { – 1, 0, 4}. Let P(x) be the predicate (x – 3)(x + 4)(x – 3)(x + 4) = 0 is false.= 0. Show that 3x E D such that

Answered: d) Let D = { – 1, 0, 4}. Let P(x) be…

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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Related questions

Question

Use truth sets to prove or disprove the following quantified statements.
Note: You can view the solutions by clicking on "Show Detailed Solution" at the end of the problem.
d) Let D = { – 1, 0, 4}. Let P(x) be the predicate (x – 3)(x + 4)
(x – 3)(x + 4) = 0 is false.
= 0. Show that 3x E D such that

Expert Solution

Step 1

A quantified statement is defined as a statement whose truth value may depend on the values of some variables.

For example: the truth value of the statement $x^{2} < 0$ depends on the value of $x$ . So the quantified statement, " $\forall x \in ℝ$ , $x^{2} < 0$ " is false, since it fails for $x = 0$ . Here, the quantifier is "for all" represented by " $\forall$ " which indicates that something is true about every element in a given set.

Similarly, the truth value of the statement $x^{2} = 0$ depends on the value of $x$ . So the quantified statement, " $\exists x \in ℝ$ , $x^{2} = 0$ " is true, since it is true for $x = 0$ . Here, the quantifier is "there exists" represented by " $\exists$ " which indicates that something is true about at least one element in a given set.