Question

Translate each of these quantifications into English and determine its truth value.

(a) \(\forall x \in {\bf{R}}\left( {{x^2} \ne - 1} \right)\)

(b) \(\exists x \in {\bf{z}}\left( {{x^2} = 2} \right)\)

(c) \(\forall x \in {\bf{z}}\left( {{x^2} > 0} \right)\)

(d) \(\exists x \in {\bf{R}}\left( {{x^2} = x} \right)\)

Short answer

(a) “The square of all real numbers is not equals to -1”.

True

(b) “There exists an integer whose square is equal to 2”.

False

(c) “The square of all integers is positive”.

False

(d) “There exists a real number whose square is equal to the real number itself”.

True

Step-by-step solution

Definitions

X is an element of Y.

Notation:\(X \in Y\)

Essential quantification:

\(\exists xP\left( x \right)\): There exists an element x in the domain such that \(P\left( x \right)\).

Universal quantification:

\(\forall xP\left( x \right)\): \(P\left( x \right)\)for all values of x in the domain.

To translate the given quantification into English and determine its truth value(a)

Here, \(\forall x \in {\bf{R}}\left( {{x^2} \ne - 1} \right)\)

The given quantification can be translated into English as:

“The square of all real numbers is not equals to -1”.

Truth value: This statement is truebecause the square of a real number is always non-negative.

To translate the given quantification into English and determine its truth value (b)

Here, \(\exists x \in {\bf{z}}\left( {{x^2} = 2} \right)\)

The given quantification can be translated into English as:

“There exists an integer whose square is equal to 2.

Truth value: This statement is false because\(x = \sqrt 2 \)and\(x = - \sqrt 2 \) are the only values for which the equation \({x^2} = 2\)holds true, but \(x = \sqrt 2 \) and\(x = - \sqrt 2 \)are not integers.

To translate the given quantification into English and determine its truth value (c)

Here, \(\forall x \in {\bf{z}}\left( {{x^2} > 0} \right)\)

The given quantification can be translated into English as:

“The square of all integers is positive”.

Truth value: This statement is false because 0 is an integer and its square is \({0^2} = 0\) . Thus,0 has a nonpositive square.

To translate the given quantification into English and determine its truth value (d)

Here, \(\exists x \in {\bf{R}}\left( {{x^2} = x} \right)\)

The given quantification can be translated into English as:

“There exists a real number whose square is equal to the real number itself”.

Truth value: This statement istruebecause \(x = 1\) has the property that it is equal to its square.