6.7. Question 6

Part a of the question is we have to verify whether the conjunction of p → r and q → r is logically equivalent to the implication p or q → r. So again you can use truth table method you can draw the truth table for that LHS part here, you can draw the truth table for this RHS part and then check whether both the truth tables are same or not.

In this section, we're discussing how to verify whether two logical expressions are equivalent. We have two expressions to compare: the left-hand side (LHS) which is p → r and q → r, and the right-hand side (RHS) which is p or q → r. The verification can be done using a truth table that lists all possible truth values of the variables p, q, and r. If the truth values of both expressions match for all possible combinations, they are logically equivalent. If they do not match, they are not equivalent.

Think of logical equivalence like two routes leading to the same destination. If you take both routes and end up at the same place regardless of the path chosen (i.e., regardless of the truth values), then those routes (or expressions) are equivalent.

We will not do that; we want to apply various identities, rules of inferences and so on. So we start with the LHS namely p → r conjunction q → r. Somehow I will try to bring it into my RHS part. So what I can do is I can replace this p → r by ¬ p or r because I know that p → q is equivalent to the disjunction of ¬ p and q.

Here we are transforming the left-hand side of the logical expression for easier comparison with the right-hand side. The expression p → r can be rewritten using the identity: p → r is the same as ¬p or r. This means that we can express the LHS as a combination of disjunctions instead of implications, which may make it easier to see how it relates to the RHS.

Imagine you have a rule that states 'If it rains, I will take an umbrella.' You can think of this as saying, 'Either it's not raining, or I have an umbrella.' This transformation allows for a clearer understanding of all possible outcomes (like you'd do with logical expressions).

Now what I can do is. I can apply the distributive law and simplify the conjunction of these two clauses I can bring it into this form. Because if indeed I apply the distributive law the disjunction goes inside and r also goes once with ¬ p and once with ¬ q.

The distributive law of logic allows us to expand or simplify logical expressions. By distributing the conjunction over the disjunction, we can rewrite the combined terms in a more manageable format. This is crucial for showing equivalence, as it helps us express the compound statements in terms of simple logical operations that we can compare directly.

Think of distributing like sharing candies among friends. If you have a bag of mixed candies (like our compound expression) and you want to make sure every friend (every term in the logic) gets their fair share according to specific rules, distributing ensures that each friend knows what they are getting (just like how we clarify our expressions).

And then again, I can apply this law namely, ¬ p or q is equivalent to p → q here. So you can imagine that, this whole thing is some s and r. So this is the form of ¬ s or r and this is equivalent to s → r and then you can substitute back s to be p or q.

De Morgan's Law provides ways to negate conjunctions and disjunctions. It states that the negation of a disjunction is equivalent to the conjunction of the negations, and vice versa. We can rewrite our expression in terms of implications again ('¬ p or q'), which allows for easier manipulation and comparison with our RHS.

Consider a scenario where you say, 'If I don’t go to the party, or I’m not tired, then I will go out.' De Morgan's law helps us analyze the conditions under which going out is decided, just like the way we manipulate and simplify expressions using logical identities.

What this is? This is nothing but your RHS that means starting with LHS by keep on simplifying it we can convert it into our RHS form and hence I can conclude that my LHS and RHS are logically equivalent.

After all transformations and manipulations, if we can reach an expression that matches the original RHS structure, we successfully demonstrate the logical equivalence of the two expressions. The manipulations we performed maintain the truth values throughout, confirming that LHS and RHS yield the same results under all possible valuations of their variables.

Imagine solving a complex algebraic equation. If you start with one side and systematically transform it through operations until it looks just like the other side, you’ve proven both sides are equal. Logical expressions are similar; when you equate their structure and truth values, you've shown they are equivalent.

The second part of this question is we have to verify whether (p → q) → r is logically equivalent to p → (q → r). So I have explicitly added the parenthesis here.

In this part, we need to check if modifying the order of operations (by adding parentheses) changes the truth values of the expressions. Here, p → q is placed inside the implication of r, which can offer a different logical structure than when p leads directly to the implication of q and r. Counterexamples can help identify these differences by showing a case where the truth values diverge for specific variable assignments.

Think of two friends deciding on plans: 'If Alex goes (p), then Charlie goes (q)' is different from 'If Alex goes, then if Charlie goes, we will see a movie (r)'. The order of decisions matters; just like in these expressions, the order can lead to different outcomes.

For the truth assignment that I have considered here, the two expressions have different truth assignments or truth values.

By assigning specific truth values (like true or false) to p, q, and r, we can demonstrate how the two different expressions yield differing results. Finding one such instance is sufficient to show that the two expressions are not logically equivalent, as they cannot hold the same truth value for all possible assignments.

Think about playing a game with rules that lead to different outcomes based on initial choices. If one player does not choose a color, the game's outcome can differ from when the color is chosen. This unpredictability in choices mirrors how logical expressions can yield different truth values based on the order and structure of their components.

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