Question

Express these system specifications using the propositions p "The message is scanned for viruses" and q "The message was sent from an unknown system" together with logical connectives (including negations).
a) "The message is scanned for viruses whenever the message was sent from an unknown system."
b) "The message was sent from an unknown system but it was not scanned for viruses."
c) "It is necessary to scan the message for viruses whenever it was sent from an unknown system."
d) "When a message is not sent from an unknown system it is not scanned for viruses."

Short answer

1. The logical proposition of the given statement is determined as $\mathbf{q}\mathbf{\to }\mathbf{p}$.
2. The logical proposition of the given statement is determined as localid="1668086079121" $\mathit{q}\mathbf{\wedge }\mathbf{~}\mathit{p}$.
3. The logical proposition of the given statement is determined as $\mathbf{q}\mathbf{\to }\mathbf{p}\mathbf{.}$
4. The logical proposition of the given statement is determined as ~q→~p.

Step-by-step solution

Introduction to the Concept

"If-then"is a logical sign. It can be depict as $\mathbf{\to }$.

“but__not” is a logical sign. It can be depict as$\mathbf{\wedge }\mathbf{~}$.

"not" is a logical sign. It can be depict as $\mathbf{-}$.

Given statements

The given statements are,

p: The message is scanned for viruses.

q: The message was sent from an unknown system.

Solution Explanation

a)

The given statement is,

The message is scanned for viruses whenever themessage was sent from an unknown system.

The logical proposition of the above statement is given as $\mathbf{q}\mathbf{\to }\mathbf{p}$.

Solution Explanation

b)

The given statement is,

The message was sent from an unknown system but it was not scanned for viruses.

The logical proposition of the above statement is given as$\mathbf{q}\mathbf{\wedge }\mathbf{~}\mathbf{p}$.

Solution Explanation

c)

The given statement is,

It is necessary to scan the message for viruses whenever it was sent from an unknown system.

The logical proposition of the above statement is given as $\mathbf{q}\mathbf{\to }\mathbf{p}$.

Solution Explanation

d)

The given statement is,

When a message is not sent from an unknown system it is not scanned for viruses.

The word "not" appears twice in this phrase, suggesting that both p and q are innegation. Another thing to note is the word "when."

This statement states unequivocally that if p is false, then q is false as well. As a result, it will be written as ~q→~p.

The logical proposition of the above statement is given as ~q→~p.