Chapter B.1, Problem 3E

Solution

(a.)

To Make: The given sentence false, using quantifiers, where $x$ is a natural number.

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x+8=11$ for all $x$ .

Given:

  $x+8=11$

Concept used:

The quantifiers ‘all’, ‘every’ and ‘no’ refer to each and every element in a set and are called universal quantifiers. The quantifiers ‘some’ and ‘there exists’ refer to one or more or possibly all elements in a set and are called existential quantifiers.

Calculation:

The given sentence is $x+8=11$ .

This is true if $x=3$ , which satisfies the fact that $x$ is a natural number; and is false otherwise.

Thus, the given sentence can be made false by using a universal quantifier as follows:

  $x+8=11$ for all $x$ .

Conclusion:

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x+8=11$ for all $x$ .

(b.)

To Make: The given sentence false, using quantifiers, where $x$ is a natural number.

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x+0=x$ for no $x$ .

Given:

  $x+0=x$

Concept used:

The quantifiers ‘all’, ‘every’ and ‘no’ refer to each and every element in a set and are called universal quantifiers. The quantifiers ‘some’ and ‘there exists’ refer to one or more or possibly all elements in a set and are called existential quantifiers.

Calculation:

The given sentence is $x+0=x$ .

This is true for any value of $x$ , provided it satisfies the fact that $x$ is a natural number.

Thus, the given sentence can be made false by using a universal quantifier as follows:

  $x+0=x$ for no $x$ .

Conclusion:

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x+0=x$ for no $x$ .

(c.)

To Make: The given sentence false, using quantifiers, where $x$ is a natural number.

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x^2=4$ for all $x$ .

Given:

  $x^2=4$

Concept used:

The quantifiers ‘all’, ‘every’ and ‘no’ refer to each and every element in a set and are called universal quantifiers. The quantifiers ‘some’ and ‘there exists’ refer to one or more or possibly all elements in a set and are called existential quantifiers.

Calculation:

The given sentence is $x^2=4$ .

This is true if $x=\pm 2$ , out of which $x=2$ satisfies the fact that $x$ is a natural number; and is false otherwise.

Thus, the given sentence can be made false by using a universal quantifier as follows:

  $x^2=4$ for all $x$ .

Conclusion:

It has been determined that the given sentence can be made false using quantifiers as follows:

  $x^2=4$ for all $x$ .

(d.)

To Make: The given sentence false, using quantifiers, where $x$ is a natural number.

It has been determined that the given sentence can be made false using quantifiers as follows:

There exists some $x$ for which $x+1=x+2$ .

Given:

  $x+1=x+2$

Concept used:

The quantifiers ‘all’, ‘every’ and ‘no’ refer to each and every element in a set and are called universal quantifiers. The quantifiers ‘some’ and ‘there exists’ refer to one or more or possibly all elements in a set and are called existential quantifiers.

Calculation:

The given sentence is $x+1=x+2$ .

This is false for any value of $x$ , such that $x$ is a natural number.

Thus, the given sentence can be made false by using an existential quantifier as follows:

There exists some $x$ for which $x+1=x+2$ .

Conclusion:

It has been determined that the given sentence can be made false using quantifiers as follows:

There exists some $x$ for which $x+1=x+2$ .