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Logical implication is a fundamental concept in logic. It is formulated as \( p \rightarrow q \). This statement means that if \( p \) is true, then \( q \) must also be true. There are key points to remember about logical implications: \( p \rightarrow q \) is equivalent to \( eg p \vee q \). This means the implication can be seen as either \( p \) is not true, or \( q \) is true.
Understanding this equivalency helps simplify logical expressions. It allows the rewriting of implications in different forms that might be easier to work with in proofs or logical evaluations.

Logical equivalence occurs when two statements, no matter how they are written, always have the same truth value. For example, \( p \rightarrow q \) is equivalent to \( eg p \vee q \).
This means, no matter what truth values \( p \) and \( q \) take, the expressions on both sides will always reflect the same truth value. As a valuable tool in logic and mathematics, logical equivalence enables us to simplify complex logical expressions into more understandable forms.
For instance, to evaluate whether \( (eg p \wedge (p \rightarrow q)) \rightarrow eg q \) is a tautology, we use logical equivalence to replace \( p \rightarrow q \) with \( eg p \vee q \).

In logic, the distributive law allows for the restructuring of logical expressions to simplify and better understand them. It states: \[ p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r) \]. This law proves valuable when working with the expression \( (eg p \wedge (eg p \vee q)) \rightarrow eg q \).
This expression can become simpler using the distributive law. By applying the distributive law, we get: \( (eg p \wedge eg p) \vee (eg p \wedge q) \rightarrow eg q \). Simplifying this using logical equivalence gives us \( eg p \rightarrow eg q \).
Remember, the goal of using the distributive law is to transform logical expressions into more manageable forms. This approach aids significantly in proving or disproving propositions and tautologies.