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Negations are essential in mathematical logic and play a crucial role in expressing statements in a clear and precise manner. A negation essentially denies the truth value of a given proposition. If a proposition P(x) is 'true,' then eg P(x) means 'P(x) is not true,' or 'P(x) is false.'

In our exercise, the domain consists of integers 1, 2, and 3, and we need to express eg P(x) for each of these values. For instance, eg P(1) means that 'P(1) is not true.'

Negations help in forming expressions that capture the true essence of the unique existential quantifier. By using negations, we ensure that none of the other values in the domain satisfy the condition P(x) when one specific value does.

Conjunctions combine multiple statements and present them as a single joint statement. In logic, this is denoted by \land, which represents the logical 'and.' For a conjunction to be true, all individual components must be true.

In our exercise, we use conjunctions to link the truth value of P(x) with the falsehood of P for other values in the domain. For instance, we write this as \(P(1) \land eg P(2) \land eg P(3)\) to mean 'P(1) is true and P(2) is not true and P(3) is not true.'

Conjunctions are integral in ensuring that all necessary conditions are met simultaneously. They help us express the idea that a particular value makes P(x) true while ensuring other values do not.

Disjunctions allow for flexibility in statements by presenting options. Denoted by \(\lor\), disjunctions represent the logical 'or'. A disjunction is true if at least one of the individual statements it connects is true.

In our exercise, disjunctions connect the different scenarios where exactly one value in the domain satisfies P(x). For example, \(P(1) \land eg P(2) \land eg P(3))\lor\ (P(2) \land eg P(1) \land eg P(3))\lor\ (P(3) \land eg P(1) \land eg P(2))\) states that either 'P(1) is true and the others are not', 'P(2) is true and the others are not,' or 'P(3) is true and the others are not.'

Disjunctions are essential for combining multiple logical possibilities into a comprehensive statement. They make it possible to express the full range of scenarios that meet the unique existence condition.