Analyze the logical forms of the following statements. The...

Analyze the logical forms of the following statements. The universe of discourse is $\mathbb{R}$. What are the free variables in each statement? (a) Every number that is larger than $x$ is larger than $y$. (b) For every number $a$, the equation $a x^2+4 x-2=0$ has at least one solution iff $a \geq-2$. (c) All solutions of the inequality $x^3-3 x<3$ are smaller than 10 . (d) If there is a number $x$ such that $x^2+5 x=w$ and there is a number $y$ such that $4-y^2=w$, then $w$ is between -10 and 10 .

Analyze the logical forms of the following statements. The universe of discourse is $\mathbb{R}$. What are the free variables in each statement?
(a) Every number that is larger than $x$ is larger than $y$.
(b) For every number $a$, the equation $a x^2+4 x-2=0$ has at least one solution iff $a \geq-2$.
(c) All solutions of the inequality $x^3-3 x<3$ are smaller than 10 .
(d) If there is a number $x$ such that $x^2+5 x=w$ and there is a number $y$ such that $4-y^2=w$, then $w$ is between -10 and 10 .

Key Concepts

Bound Variables

Bound variables are those that are quantified within a logical expression by operators like '?' (for all) or '?' (there exists). They have a local meaning within the scope of the quantifier, and any changes to their names do not affect the overall meaning of the expression. Recognizing bound variables helps in understanding the structure of logical formulas.

Quantifiers

Quantifiers are logical operators that specify the number of elements in the universe of discourse for which a statement holds true. The most common are the universal quantifier (?) and the existential quantifier (?). They dictate whether the variables in the expression are free or bound, and thus play a key role in the logical structure of the statement.

Universe of Discourse

The universe of discourse is the set of objects over which all variables in a logical statement range. In our context, all variables are assumed to be real numbers, which dictates the domain for interpreting quantitative statements.

Free Variables

Free variables in a logical expression are those that are not quantified by any logical operator such as 'for all' or 'there exists'. Their values remain unspecified, making the statement open-ended. The identification of free variables is crucial for understanding which variables can be arbitrarily assigned values from the universe of discourse.

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