Let L(x, y) be the propositional function "x loves y." The...

Let $L(x, y)$ be the propositional function "x loves y." The domain of discourse is the Cartesian product of the set of all living people with itself (i.e., both $x$ and $y$ take on values in the set of all living people). Write each proposition symbolically. Which do you think are true? Everybody loves somebody.

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Key Concepts

Domain of Discourse

The domain of discourse specifies the set of elements over which the quantifiers range. In this instance, it is the Cartesian product of the set of all living people, meaning that both x and y represent living individuals.

Predicate Functions

Predicate functions assign a truth value to a statement based on the properties of or relationships between elements in the domain. In this context, L(x, y) is a predicate that indicates whether one person loves another, defining relationships within the set of all living people.

Universal Quantifier

This represents the concept of 'every' or 'all' within predicate logic. It is used to assert that a statement holds for every element in the domain. In the given problem, it applies to 'everybody', meaning each individual referenced in the statement.

Existential Quantifier

This embodies the idea of 'there exists' or 'some' in formal logic. It's used to claim that at least one element in the domain satisfies a given condition. In the problem, it applies to 'somebody', indicating that for each person there is at least one person that they love.

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