Question

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.

Short answer

1. The proposition is $\boxed{\neg p}$.
2. The proposition is$\boxed{p\wedge \neg q}$.
3. The proposition is$\boxed{p\to q}$.
4. The proposition is$\boxed{\neg p\to \neg q}$.
5. The proposition is$\boxed{p\to q}$.
6. The proposition is $\boxed{q\wedge \neg p}$.
7. The proposition is $\boxed{q\to p}$.

Step-by-step solution

Introduction

A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false".

Consider the following proposition,

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

$q:$You get a speeding ticket.

The objective is to write the following propositions using $p$and $q$.

(a) Write the propositions using p and q

Consider the statement as,

You do not drive over 65 miles per hour.

The equivalentpropositions to the above statement in terms of p and q is,

$\boxed{\neg p}$.

Hence, the proposition is $\boxed{\neg p}$.

(b) Write the propositions using p and q

Consider the statement as,

“You drive over 65 miles per hour but you do not get a speeding ticket”.

The equivalent proposition to the above statement in terms of p and q is,

$\boxed{p\wedge \neg q}$.

Hence, the proposition is $\boxed{p\wedge \neg q}$.

(c) Write the propositions using p and q

Consider the statement as, “You will get a speeding ticket if you drive over 65 miles per hour”.

The equivalent proposition to the above statement in terms of p and q is

$\boxed{p\to q}$.

Hence, the proposition is $\boxed{p\to q}$.

(d) Write the propositions using p and q

Consider the statement as,

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

The equivalent proposition to the above statement in terms of p and q is $\boxed{\neg p\to \neg q}$.

Hence, the proposition is $\boxed{\neg p\to \neg q}$.

(e) Write the propositions using p and q

Consider the statement as,

“Driving over 65 miles per hour is sufficient for getting a speeding ticket”.

The equivalent proposition to the above statement in terms of p and q is, $\boxed{p\to q}$.

Hence, the proposition is $\boxed{p\to q}$.

(f) Write the propositions using p and q

Consider the statement as,

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

The equivalent proposition to the above statement in terms of p and q is$\boxed{q\wedge \neg p}$.

Hence, the proposition is $\boxed{q\wedge \neg p}$.

(g) Write the propositions using p and q

Consider the statement as,

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

The equivalent proposition to the above statement in terms of p and q is,

$\boxed{q\to p}$.

Hence, the proposition is $\boxed{q\to p}$.