7. In order to disprove the implication that P implies Q, one often provides an example in which P is true but Q is not. Such an example is called a counterexample to the statement that P implies Q. For each of the following incorrect statements, identify P, Q, and provide a counterexample. (i) If an integer is divisible by 3 then it is divisible by 9. (ii) All quadratic polynomials have two real roots (iii) If a function f from R to R is one-to-one, then the function f² is one-to-one (iv) If a function f from R to R is one-to-one and bounded, then f-1 is one-to-one and bounded.
7. In order to disprove the implication that P implies Q, one often provides an example in which P is true but Q is not. Such an example is called a counterexample to the statement that P implies Q. For each of the following incorrect statements, identify P, Q, and provide a counterexample. (i) If an integer is divisible by 3 then it is divisible by 9. (ii) All quadratic polynomials have two real roots (iii) If a function f from R to R is one-to-one, then the function f² is one-to-one (iv) If a function f from R to R is one-to-one and bounded, then f-1 is one-to-one and bounded.
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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Transcribed Image Text:7. In order to disprove the implication that P implies Q, one often provides an example in which P
is true but Q is not. Such an example is called a counterexample to the statement that P implies
Q. For each of the following incorrect statements, identify P, Q, and provide a counterexample.
(i) If an integer is divisible by 3 then it is divisible by 9.
(ii) All quadratic polynomials have two real roots
(iii) If a function f from R to R is one-to-one, then the function f2 is one-to-one
(iv) If a function f from R to R is one-to-one and bounded, then f-1 is one-to-one and bounded.
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