Problem 26E

The quantifier ∃n denotes "there exists exactly n,” so that means there exist exactly n values in the domain such that P(x) is true. Determine the true value of these statements where the domain consists of all real numbers.

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.