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Compound propositions are logical statements formed by combining other propositions using logical connectives such as 'and' (\( \land \)), 'or' (\( \lor \)), 'not' (\( eg \)), and 'if...then' (\( \rightarrow \)). Each component of a compound proposition is also a proposition. For instance, if we have two propositions \(p\) and \(q\), we can form a compound proposition like \(p \land q\), which reads as 'p and q'.

A compound proposition differs from a basic proposition because it can express more complex logical statements. The truth value of a compound proposition depends on the truth values of its components and the logical connectives used.

Understanding compound propositions is crucial in logic as they allow us to express and analyze complex ideas. They are used in various fields, such as computer science, mathematics, and philosophy, to reason about conditions, make decisions, and form valid arguments. In the given exercise, the propositions \(p\), \(q\), and \(r\) are compound propositions, meaning they are built from simpler statements using logical connectives.

A tautology is a logical statement that is always true, regardless of the truth values of its component propositions. In other words, a tautology is a proposition that cannot be false in any scenario. For example, the proposition \(p \lor eg p\) is a tautology because it covers all possible truth values of \(p\); whether \(p\) is true or false, the statement will always be true.

The importance of tautologies in logic cannot be overstated. They serve as fundamental truths that help in proving other logical statements. In the given exercise, the logical equivalence of propositions is demonstrated using tautologies.

Specifically, to show that \(p\) and \(q\) are logically equivalent (\(p \equiv q\)), we need to prove that \(p \leftrightarrow q\) is a tautology. This means that the bi-conditional statement \(p \leftrightarrow q\) must be true for all possible truth values of \(p\) and \(q\). Similarly, the logical equivalence \(q \equiv r\) is shown by proving that \(q \leftrightarrow r\) is a tautology. Finally, the transitivity of these tautologies helps us deduce that \(p \leftrightarrow r\) is also a tautology, thereby proving \(p \equiv r\).

Transitivity is a property that applies to logical equivalence among propositions. If a relation is transitive, it means that if one element is related to a second, and the second is related to a third, then the first element is also related to the third. In terms of logical equivalence, transitivity can be stated as: if \(p \equiv q\) and \(q \equiv r\), then \(p \equiv r\).

This property is essential for connecting chains of logical equivalences. If we know two propositions are equivalent to a common third proposition, then they must be equivalent to each other. This principle is used in the given exercise to show that if \(p\) is equivalent to \(q\) and \(q\) is equivalent to \(r\), then \(p\) must be equivalent to \(r\).

The exercise utilizes transitivity as follows:

• We start with the given equivalences: \(p \equiv q\) and \(q \equiv r\).
• Since \(q\) serves as a common link between \(p\) and \(r\), we can apply transitivity to conclude that \(p \equiv r\).

Therefore, transitivity provides a powerful method to deduce relationships between propositions, forming the backbone of equivalence proofs in logic.