6.1. Discrete Mathematics

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.

Propositional variables are symbols used to represent statements that can be either true or false. In this case, 'p' represents the statement 'you drive over 65 miles per hour', and 'q' represents 'you get a speeding ticket'. Understanding what these variables mean is crucial because they will help us construct logical expressions based on them.

Think of propositional variables like labels on jars. Just as a jar labeled 'cookies' means it contains cookies, the variable 'p' tells us about a specific action (driving speed). Understanding what each label means helps us know what we have.

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.

To represent the statement 'you do not drive over 65 miles per hour', we use the negation symbol (¬). Since 'p' means 'you drive over 65 miles per hour', its negation '¬p' means the opposite – that you are driving below or equal to 65 miles per hour. This indicates that negation is a way to express the opposite of a given proposition.

Imagine you're at a party, and someone says they are not going to dance (¬d). If 'd' represents dancing, then ¬d means they are opting out of dancing, clearly showing an alternative state.

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.

This statement is written in an if-then form, which is represented as 'p → q'. Here, 'if p' is the condition, and 'q' is the conclusion that follows. This means that driving over 65 miles per hour (p) results in getting a speeding ticket (q). The implication 'p → q' signifies that the truth of p guarantees the truth of q.

Think of a teacher saying, 'If you study hard (p), you will pass the exam (q)'. This means studying hard is a condition for passing, showing the direct relationship between the action (studying) and the outcome (passing).

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. This is a statement of the form only if, so recall p → q also represents p only if q.

The phrase 'only if' is crucial because it establishes a necessary condition. Here, driving over 65 miles per hour (p) can only occur if you get a speeding ticket (q). This can be expressed as 'p → q' or in its contrapositive form '¬q → ¬p', meaning if you do not get a speeding ticket, then you did not drive over that speed.

Consider a scenario where you tell a friend, 'You can have dessert only if you finish your vegetables.' Here, finishing the vegetables is a necessary condition for having dessert, illustrating direct dependency.

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 thereafte,r sufficient that will happen.

When we say that driving over 65 miles per hour is 'sufficient' for getting a speeding ticket, it indicates that this action alone is enough to guarantee the outcome. This is also represented as 'p → q', illustrating a direct cause-effect relationship where the action of speeding results directly in receiving a ticket.

If you think of a rule in a game: 'Scoring three points is sufficient to win'. Here, scoring three points secures the win; you don't need to score more. It highlights how a certain condition alone can lead to a definite outcome.

In question 2 the goal is the following, you have to write down the converse contrapositive and inverse of the following statements. ... So just to recall, if you are given an implication p → q then the contrapositive of that is ¬q → ¬p.

In logic, for any statement expressed as an implication 'p → q', we can derive three related statements: the converse (q → p), the inverse (¬p → ¬q), and the contrapositive (¬q → ¬p). These transformations help analyze the relationships between conditions and their outcomes, providing alternative angles to the original implication.

Think of a traffic rule: 'If it rains, the roads are wet' (p → q). The converse (roads are wet → it rains) might not always hold true, such as when there are water leaks. The inverse and contrapositive provide more context, allowing us to question the reliability of our conclusions based on the truth of our initial premise.

The question 3 is asking you to do the following. You are given a set of compound propositions and you have to draw the truth table it is a very straightforward question here.

Truth tables are used to evaluate the truth values of compound statements formed from logical connectives like 'and', 'or', and 'not'. By systematically organizing the potential truth values of the variables involved, you can determine the overall truth of complex expressions based on simple ones.

Imagine a recipe that uses certain ingredients. A truth table is like a checklist showing whether each ingredient is available. If all required ingredients are 'true' (available), then the dish can be made; if any are 'false' (not available), it's clear that the dish can't come together.

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