e. Then write the negation of the statement in Engli -ls for quantifiers. VmeZ) (m² is even). VxR) (x² > 0). or each real number x, √x € R. m Vm=Z) (77 € Z). 3 va € Z) (√a² = a). \x € R) (tan² x + 1 = sec² x).

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
Problem 20E: Give counterexamples for the following statements. If and are irrational, then is irrational. If ...
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2. For each of the following, use a counterexample to show that the statement
is false. Then write the negation of the statement in English, without using
symbols for quantifiers.
(a) (Vm = Z) (m² is even).
★
(b) (Vx € R) (x² > 0).
(c) For each real number x, √√x € R.
m
(d) (Vm € Z) (€Z).
3
(e) (va € Z) (√a² = a).
*
(f) (Vx € R) (tan² x + 1 = sec² x).
Transcribed Image Text:2. For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers. (a) (Vm = Z) (m² is even). ★ (b) (Vx € R) (x² > 0). (c) For each real number x, √√x € R. m (d) (Vm € Z) (€Z). 3 (e) (va € Z) (√a² = a). * (f) (Vx € R) (tan² x + 1 = sec² x).
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I believe 2f is wrong, suppose to prove that it is a false (per the question) and provide counter example.

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