6.7.2. Part B

Constructing Compound Propositions

The first statement that we want to represent here is you do not drive over 65 miles per hour, which is very simple this statement is nothing but negation of p because p represents the statement you drive over 65 miles per hour, so you want to represent the negation of that. The second statement is that you want to represent here is you will get a speeding ticket if you drive over 65 miles per hour, so this is some form of if-then statement. This is the if part this is the conclusion. So the if part here is if you drive over 65 miles per hour which is p and the conclusion here is you will get a speeding ticket. That is why this statement will be represented by p → q.

In propositional logic, we often use variables to represent statements. Here, 'p' means 'you drive over 65 miles per hour' and 'q' means 'you will get a speeding ticket'. To represent the statement 'you do not drive over 65 miles per hour', we use the negation of p, which is denoted ¬p. This acts as a way of confirming that the event does not happen. For the second statement, 'you will get a speeding ticket if you drive over 65 miles per hour', we can express that with the implication p → q. This means that if 'p' is true (you drive over the speed limit) then 'q' will also be true (you get a speeding ticket). This structure helps us to build complex logical relationships using basic statements.

Imagine you have a friend who loves to drive fast. You tell him that if he speeds (p), he will get a ticket (q). If he chooses not to speed (¬p), he's safe from tickets. This is similar to how we connect cause and effect in daily decisions, where one action directly influences the result.

The third statement that we want to represent here is you drive over 65 miles per hour only if you will get a speeding ticket. So this is a statement of the form only if, so recall p → q also represents p only if q, or equivalently q is necessary for p. These are the various forms for p → q. So this condition is the necessity condition here.

The phrase 'you drive over 65 miles per hour only if you will get a speeding ticket', indicates that getting a speeding ticket (q) is a necessary condition for speeding (p). In logic, this can also be expressed as p → q, suggesting that 'p' cannot occur without 'q' being true. In alternative logical forms, we can also express it as ¬q → ¬p, denoting that if you do not get a ticket, then you must not be speeding.

Consider a situation where a student can only pass a class (p) if they attend at least 80% of the lectures (q). This forms a necessary condition where attending enough classes is a prerequisite to passing. If the student does not pass, it is implied they must not have attended the necessary number of lectures.

The last statement we want to represent here is driving over 65 miles per hour is sufficient for getting a speeding ticket. That means whatever is there before your sufficient part that is your if statement. If you ensure that then whatever is thereafter sufficient that will happen. So this is equivalent to p → q.

The phrase 'driving over 65 miles per hour is sufficient for getting a speeding ticket' implies that if you speed (p), then you will definitely get a speeding ticket (q). This is a straightforward logical implication expressed as p → q. It shows that speeding guarantees a ticket, without any additional conditions or qualifications.

Think about how in many workplaces, showing up on time (p) is sufficient for being considered reliable (q). If an employee is consistently punctual, their reliability as a worker is unquestionable. This highlights how one action can directly ensure a certain outcome, similar to how speeding guarantees a ticket.

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