6.7.1. Part A

In this question, the goal is the following. You are given two propositional variables p and q representing the propositions you drive over 65 miles per hour and you get a speeding ticket respectively. Then your goal is the following: using these two propositions p and q, you have to represent the following statements as compound propositions using logical connectives.

In logical reasoning, propositions are statements that can either be true or false. Here, we are working with two propositional variables: 'p' which indicates the action of driving over 65 miles per hour, and 'q' which indicates whether you will get a speeding ticket. The goal is to take real-world statements and express them using these variables and logical connectives (like 'and', 'or', 'not', and 'if-then'). This process helps us analyze and understand these statements logically.

Imagine you're setting rules for playing a game. Each rule can be considered a statement that you can check (whether it's true or false). For instance, 'If you roll a six, you get an extra turn' is a rule in logical terms that can be written as 'if p, then q.'

The first statement that we want to represent here is: you do not drive over 65 miles per hour. This statement is represented as ¬p, which is the negation of p because p represents the statement you drive over 65 miles per hour.

The negation of a statement is the opposite of that statement. In this case, the original statement is 'you drive over 65 miles per hour.' To represent the opposite, we use ¬p, denoting 'not p', which clearly indicates that the driver is not speeding.

Think of it like making a promise. If you promise to be home by 8 PM and say 'I will not be home by 8 PM,' it clearly states the opposite of what you promised.

The second statement is that you will get a speeding ticket if you drive over 65 miles per hour. This is represented by p → q, where p is the condition (driving over 65) and q is the outcome (getting a ticket).

This statement is an example of a conditional statement or implication. It indicates that whenever the condition is met (you drive over 65 mph), then the consequence follows (you get a ticket). If p is true and leads to q being true, then the implication holds.

Consider the scenario of watering a plant. If you water the plant (p), then it will thrive (q). Saying 'if p, then q' creates a direct relationship between the action and its outcome.

The third statement we want to represent is: you drive over 65 miles per hour only if you will get a speeding ticket. This can also be expressed as p → q and is the necessity condition.

This phrase 'only if' indicates that the occurrence of p is conditional on q being true. This does not change the representation to p → q but highlights the relationship where p cannot occur without q. It implies that if you’re speeding (p), then it is necessary that you receive a speeding ticket (q).

Think of it like needing a ticket to get into a concert. You cannot enter (p) without having a ticket (q). Thus, entering only happens if you have the ticket.

The last statement is: driving over 65 miles per hour is sufficient for getting a speeding ticket. This can be again represented as p → q.

In logic, saying that p is sufficient for q means that if p happens, then q will necessarily happen. This reinforces our understanding of the cause-and-effect relationship in the logical context that leads to a speeding ticket whenever maximum speed is exceeded.

Imagine a situation where studying (p) is sufficient for passing an exam (q). If you study, it guarantees you will pass. Thus, studying directly leads to passing, similar to speeding leading to a ticket.

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