Question

Q : Let p and q be the propositions
p : You drive over 65 miles per hour.
q : You get a speeding ticket.
Write these propositions using p and q and logical connectives (including
negations).

a) You do not drive over 65 miles per hour.
b) You drive over 65 miles per hour, but you do not get a speeding ticket.
c) You will get a speeding ticket if you drive over 65 miles per hour.
d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.
e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.
f) You get a speeding ticket, but you do not drive over 65 miles per hour.
g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

Answer 1

ANSWER on Question #74711 – Math – Discrete Mathematics

QUESTION

Let $p$ and $q$ be the proposition

$p$ : You drive over 65 miles per hour.

$q$ : You get a speeding ticket.

Write these propositions using $p$ and $q$ and logical connectives (including negations).

a) You do not drive over 65 miles per hour.

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

c) You will get a speeding ticket if you drive over 65 miles per hour.

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.

e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

SOLUTION

a) You do not drive over 65 miles per hour.

This sentence is a negative to the sentence: "You drive over 65 miles per hour".

Conclusion, in a symbolic form this sentence looks like this

> You do not drive over 65 miles per hour: $\neg p$

b) You drive over 65 miles per hour, but you do not get a speeding ticket.

The sentence consists of two parts:

1) You drive over 65 miles per hour;

2) You do not get a speeding ticket.

This sentence is a negative to the sentence: "You get a speeding ticket".

Conclusion, in a symbolic form this sentence looks like this

You do not get a speeding ticket : ¬q

General conclusion, in a symbolic form initial sentence looks like this

You drive over 65 miles per hour, but you do not get a speeding ticket : (p∧¬q)

c) You will get a speeding ticket if you drive over 65 miles per hour.

The sentence consists of two parts:

1) You drive over 65 miles per hour – “condition”;

2) You will get a speeding ticket – “the result of the condition”.

Conclusion, in a symbolic form this sentence looks like this

You will get a speeding ticket if you drive over 65 miles per hour : (p → q)

d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket

The sentence consists of two parts:

1) If you do not drive over 65 miles per hour - "condition";

The sentence itself is a negative to the sentence: "You drive over 65 miles per hour".

2) you will not get a speeding ticket - "the result of the condition";

The sentence itself is a negative to the sentence: "You get a speeding ticket".

Conclusion, in a symbolic form this sentence looks like this

If you do not drive over 65 miles per hour, then you will not get a speeding ticket :

$$\boxed{(\neg p \rightarrow \neg q)}$$

e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.

The sentence consists of two parts:

1) Driving over 65 miles per hour – "condition";

2) Sufficient for getting a speeding ticket – "the result of the condition".

Conclusion, in a symbolic form this sentence looks like this

Driving over 65 miles per hour is sufficient for getting a speeding ticket: $(p \rightarrow q)$

f) You get a speeding ticket, but you do not drive over 65 miles per hour.

The sentence consists of two parts:

1) You get a speeding ticket;

2) You do not drive over 65 miles per hour.

This sentence is a negative to the sentence: "You drive over 65 miles per hour".

Conclusion, in a symbolic form this sentence looks like this

You do not get a speeding ticket : ¬p

General conclusion, in a symbolic form initial sentence looks like this

You get a speeding ticket, but you do not drive over 65 miles per hour : (q ∧ ¬p)

g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.

The sentence consists of two parts:

1) Whenever you get a speeding ticket – “condition”;

2) You are driving over 65 miles per hour – “the result of the condition”.

Conclusion, in a symbolic form this sentence looks like this

You will get a speeding ticket if you drive over 65 miles per hour : (q → p)

Answer provided by https://www.AssignmentExpert.com

ANSWER:

a) $\neg p$

b) $p \wedge \neg q$

c) $p \to q$

d) $\neg p \to \neg q$

e) $p \to q$

f) $q \wedge \neg p$

g) $q \to p$