Chapter B.2, Problem 1E

Solution

(a.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $p\to q$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “If it is raining, then the grass is wet.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “If it is raining, then the grass is wet.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $p\to q$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $p\to q$ .

(b.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $\sim p\to q$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “If it is not raining, then the grass is wet.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “If it is not raining, then the grass is wet.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $\sim p\to q$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $\sim p\to q$ .

(c.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $p\to \sim q$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “If it is raining, then the grass is not wet.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “If it is raining, then the grass is not wet.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $p\to \sim q$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $p\to \sim q$ .

(d.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $p\to q$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “The grass is wet if it is raining.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “The grass is wet if it is raining.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $p\to q$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $p\to q$ .

(e.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $\sim q\to \sim p$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “The grass is not wet implies that it is not raining.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “The grass is not wet implies that it is not raining.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $\sim q\to \sim p$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $\sim q\to \sim p$ .

(f.)

The given statement in symbolic form.

It has been determined that the given statement can be written in symbolic form as $q\leftrightarrow p$ .

Given:

  $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

The statement to be written in symbolic form is “The grass is wet if, and only if, it is raining.”

Concept used:

A compound statement can be written in symbolic form, by denoting each simple statement by a variable and using logical symbols.

Calculation:

The given statement is “The grass is wet if, and only if, it is raining.”

It is given that $p$ is the statement “It is raining” and $q$ is the statement “The grass is wet”.

Then, the given statement can be written in symbolic form as $q\leftrightarrow p$ .

Conclusion:

It has been determined that the given statement can be written in symbolic form as $q\leftrightarrow p$ .