Chapter B.1, Problem 1E

Solution

(a.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is a statement, which is false.

Given:

  $2+4=8$

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is $2+4=8$ .

This is clearly a sentence, which is false, as $2+4=6$ .

Hence, the given is a statement, which is false.

Conclusion:

It has been determined that the given is a statement, which is false.

(b.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

Shut the window.

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “Shut the window.”

This is clearly a sentence, but it has no truth value. That is, it is neither true nor false.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(c.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

Los Angeles is a state. He is in town.

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “Los Angeles is a state. He is in town.”

This is clearly not a sentence. More specifically, the given is a collection of two separate sentences, each of which has its own truth value.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(d.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

What time is it?

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “What time is it?”

This is clearly not a sentence. More specifically, it is a question.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(e.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

  $5x=15$

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “ $5x=15$ .”

This is clearly a sentence, but its truth value cannot be determined without more information. More specifically, the given sentence is true if $x=3$ and false otherwise.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(f.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is a statement, which is true.

Given:

  $3\cdot 2=6$

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “ $3\cdot 2=6$ .”

This is clearly a sentence, which is true.

Hence, the given is a statement, which is true.

Conclusion:

It has been determined that the given is a statement, which is true.

(g.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

  $2x^2>x$

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “ $2x^2>x$ .”

This is clearly a sentence, but its truth value cannot be determined without more information. More specifically, the given sentence is true if $x<0$ or $x>\frac{1}{2}$ and is false otherwise.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(h.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

This statement is false.

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “This statement is false.”

This is clearly a sentence, but its truth value cannot be determined without more information. More specifically, the given sentence is true if the statement it is referring to, is false and is false otherwise.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.

(i.)

Whether the given is a statement and whether it is true or false, if so.

It has been determined that the given is not a statement.

Given:

Stay put!

Concept used:

A statement is a sentence that is either true or false, but not both.

Calculation:

The given is “Stay put!”

This is clearly a sentence, but it has no truth value. That is, this sentence is neither true nor false.

Hence, the given is not a statement.

Conclusion:

It has been determined that the given is not a statement.