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Universal quantification is used to state that a property or statement applies to all elements within a certain domain. It's denoted by the symbol \(\forall\), which reads as 'for all'. For example, the statement \( \forall x (x > 0) \) means 'for all x, x is greater than 0'.
In the previous exercise, we looked at the equivalence \( \forall x (A \rightarrow P(x)) \). Here, it indicates that for any value of \( x\), if \( A \) is true, then \( P(x) \) must also be true.
The equivalence \( A \rightarrow \forall x P(x) \) states that if \( A \) is true, then \( P(x) \) is true for every \( x \).
By analyzing such statements, we see how universal quantification helps us formalize general truths in logical expressions.

Existential quantification is expressed using the symbol \( \exists \), meaning 'there exists'. This form of quantification asserts that there is at least one element for which the statement holds true. For example, the statement \( \exists x (x > 0) \) means 'there exists an \( x \) such that \( x \) is greater than 0'.
In the exercise, we dealt with \( \exists x(A \rightarrow P(x)) \). This means there is some \( x \) for which the implication \( A \rightarrow P(x) \) holds true.
The equivalence \( A \rightarrow \exists x P(x) \) states that if \( A \) is true, then there must be at least one \( x \) where \( P(x) \) is true. Understanding existential quantification is crucial in scenarios where proving just one instance is sufficient to validate a statement.

Logical equivalences are fundamental in simplifying and understanding logical statements. Two statements are logically equivalent if they always have the same truth value.
In the exercise, we established logical equivalencies for quantifiers. The statements \( \forall x (A \rightarrow P(x)) \equiv A \rightarrow \forall x P(x) \) and \( \exists x (A \rightarrow P(x)) \equiv A \rightarrow \exists x P(x) \) were shown to be logically equivalent.
Analyzing both forms, we demonstrated that the implications hold under the same truth conditions. Logical equivalences allow us to rewrite statements in a more convenient form without changing their meaning; they are vital in proofs and problem-solving.

A free variable in a logical formula is a variable that is not bound by a quantifier within that formula. It stands in contrast to a bound variable, which is declared by quantifiers like \( \forall \) or \( \exists \). For example, in the formula \( P(x) \), \( x \) is a free variable.
In the original exercise, it was pointed out that \( x \) does not appear as a free variable in \( A \). This fact simplifies the logical analysis since the value of \( x \) does not affect the truth value of \( A \).
Understanding the distinction between free and bound variables is essential in logic as it influences validity and scope of logical statements.