Let A={1,2, …, 9,10}. Consider each of the following...

Let $A=\{1,2, \ldots, 9,10\}$. Consider each of the following sentences. If it is a statement, then determine its truth value. If it is a propositional function, determine its truth set. (a) $(\forall x \in A)(3 y \in A)(x+y<14)$ (c) $\quad(\forall x \in A)(\forall y \in A)(x+y<14)$ (b) $\quad(\forall y \in A)(x+y<14)$ (d) $\quad(\exists y \in A)(x+y<14)$

Let $A=\{1,2, \ldots, 9,10\}$. Consider each of the following sentences. If it is a statement, then determine its truth value. If it is a propositional function, determine its truth set.
(a) $(\forall x \in A)(3 y \in A)(x+y<14)$
(c) $\quad(\forall x \in A)(\forall y \in A)(x+y<14)$
(b) $\quad(\forall y \in A)(x+y<14)$
(d) $\quad(\exists y \in A)(x+y<14)$

Key Concepts

Statements vs. Propositional Functions

This concept differentiates between a statement, which has a definite truth value (true or false), and a propositional function, which contains one or more free variables and thus does not have a truth value until those variables are specified. In the context of the problem, some parts are complete statements (all variables bound by quantifiers) while others are propositional functions awaiting assignment to the free variables.

Universal and Existential Quantifiers

Quantifiers are symbols used to indicate the extent to which a predicate is true over a set of elements. The universal quantifier (?) asserts that a statement holds for every element of the domain, while the existential quantifier (?) claims that there is at least one element in the domain for which the statement is true. Understanding these is key when evaluating or determining the truth set of logical formulas.

Free and Bound Variables

A bound variable is one that falls under the scope of a quantifier, and its value is determined by that quantifier over the domain; a free variable is not governed by any quantifier within the statement and can result in a propositional function rather than a true statement. In the problem, whether the variable x or y is free or bound impacts whether the expression is a complete statement with a truth value or a function that defines a set of values making it true.

Truth Value and Truth Set

A truth value is the attribute of a statement being either true or false, determined once all variables are quantified. A truth set, on the other hand, is the collection of all elements from the domain that satisfy a propositional function when substituted into its free variables. This concept is applied in the problem when one is required to decide the truth value of fully quantified sentences or find the truth set for expressions with free variables.

Domain of Discourse

The domain of discourse is the set of all objects under consideration for a given logical expression. In this problem, the domain is the set A = {1, 2, …, 10}. The constraints (i.e., quantifiers and conditions like x+y<14) are evaluated with respect to this domain.

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