Questions: p, q, and r represent the following simple statements. p: This is an octopus. q: The aquarium is full of fish. r: There are penguins. Write the following compound statement in its symbolic form. If this is not an octopus then: the aquarium is full of fish or there are penguins.

p, q, and r represent the following simple statements.
p: This is an octopus.
q: The aquarium is full of fish.
r: There are penguins.
Write the following compound statement in its symbolic form.
If this is not an octopus then: the aquarium is full of fish or there are penguins.

Transcript text: Let $p, q$, and $r$ represent the following simple statements. p : This is an octopus. q : The aquarium is full of fish. $r$ : There are penguins. Write the following compound statement in its symbolic form. If this is not an octopus then: the aquarium is full of fish or there are penguins.

Solution

Solution Steps

To convert the given compound statement into its symbolic form, we need to identify the logical connectors and the negation involved. The statement "If this is not an octopus then: the aquarium is full of fish or there are penguins" can be broken down as follows:

"This is not an octopus" is the negation of $ p $, which is $ \neg p $.
"The aquarium is full of fish or there are penguins" is $ q \lor r $.
The entire statement is a conditional statement, which can be written as $ \neg p \rightarrow (q \lor r) $.

Step 1: Identify the Negation

The statement "This is not an octopus" is the negation of $ p $. In symbolic form, this is represented as $ \neg p $.

Step 2: Identify the Disjunction

The statement "The aquarium is full of fish or there are penguins" is a disjunction of $ q $ and $ r $. In symbolic form, this is represented as $ q \lor r $.

Step 3: Form the Conditional Statement

The entire compound statement is a conditional statement: "If this is not an octopus then: the aquarium is full of fish or there are penguins." In symbolic form, this is represented as $ \neg p \rightarrow (q \lor r) $.