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The quantifier n denotes "there exists exactly n,” so that

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 26E Chapter 1.SE

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 26E

The quantifier  denotes “there exists exactly ,” so that means there exist exactly  values in the domain such that  is true. Determine the true value of these statements where the domain consists of all real numbers.

a)                  b) 

c)                  d) 

Step-by-Step Solution:

Solution :

Step 1:

In this problem, we need to determine the true value of these given statements.

Where we are given means there exists exactly n values in the domain such that P(x) is true and the domain consists of all real numbers.

(a): 0x (x2 = -1).

According to the given condition 0x (x2 = -1) means there exists exactly zero(0) value in the domain.

Clearly x2 = -1, is only possible in the complex numbers because x = is complex value.

Which is called i(iota).

Therefore, there is no value in the real numbers which is held this condition.

Hence the statement is true.

(b): 1x()

According to the given condition 1x()  means there exists exactly one(1) value in the domain.

So, we have one value zero for the condition = 0, which is held.

Hence the statement is true.

(c): 2x (x2 = 2)

According to the given condition 2x (x2 = 2)  means there exists exactly two(2) value in the domain.

Clearly x2 =2 is only possible for the real number x = +and x = - .

So, we have to real values (+) and (- ) for x, which exists for this condition x2 = 2.

Hence the statement is true.

(d): 3x (x = )

According to the given condition 3x (x = ) means there exists exactly three(3) values in the domain.

Let x = a is any real number then we get  that means  is lie between - a < x < a.

So, we have three values for this condition x = , which is - a, a and  + a.

Hence the statement is true.

Step 2 of 1

Chapter 1.SE, Problem 26E is Solved
Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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The quantifier n denotes "there exists exactly n,” so that