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)

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.