I was reading a short note about the logic when it comes to mathematical arguments. The definition is as follows,
A statement containing a free variable is called a predicate about the elements in a given set.
I am unsure how to understand this definition. I even tried searching about it on websites. Since the definition varies on some websites, I would like to ask here.
The part that I do not understand is, "about the elements in a given set". Does "about the elements" mean "for all elements"? I know well what a free variable is. Suppose that you have an expression $x+1=2$. Here, $x$ is a free variable, as it can be anything. But the $x$ only makes when talking about the numbers here, otherwise it is meaningless. So the set can be $\emptyset$, $\{2\}$, or $\mathbb{Z}$ etc. Is that what the definition says about "a given set"?
Somewhere in the note says that the expression where the quantifers is used in front of the predicate, such as $\forall x:p(x)$ is not a predicate. If you decide proving $n!<(n/2)^n$ for $n\geq 6$ by induction, how would you use "predicate" and "set" in a sentence? Which of them sounds better?
- Let $p(n)$ be a predicate for all $n\geq 6$ that stands for $n!<(n/2)^n$
- Let $p(n)$ be a predicate defined for all $n\geq 6$ that stands for $n!<(n/2)^n$
(What I mean is, that the predicate $p(n)$ is given where $n$ is desired to be natural numbers $\geq 6$.) Is there a better way to say?
When we define what it means for a first-order formula to be true or false, we refer to a structure. The structure consists of a domain and an interpretation of constant, function, and relation symbols. If the formula contains free variables, establishing its truth also requires choosing the values of the free variables.
This is the standard setup, in which quantifiers range over the domain of the structure. Let's consider your example, $x + 1 = 2$. A structure $S$ for this formula must define an interpretation for $+$, $1$, $2$, and even $=$. (Though we usually agree to give $=$ its standard interpretation.)
The structure $S$ could be the natural numbers, the rational numbers, the real numbers, the complex numbers, the integers modulo $3$, etc. with the usual interpretations of $+$, $1$, and $2$.
Let's start from $\forall x \,.\,x + 1 = 2$. We call this formula a sentence because it has no free variable occurrences. Clearly, this sentence is false in $S = \mathbb{N}$. On the other hand, $\exists x \,.\,x + 1 = 2$ is true in the same $S$.
Without quantifier, $x+1=2$ is true for some values of $x$ (exactly one value if $S = \mathbb{N}$) and false for the other values. Then, our formula, for each structure, identifies the subset of the structure's domain whose members make the formula true when assigned to $x$. That is, once we fix the structure $S$ with domain $D$, the formula with one free variables $\phi(x)$ defines the set $$ \{\, x \in D \mid \phi(x) \,\} \enspace. $$
For your example, a simple way would be to define your predicate as
$$ p(n) := n \geq 6 \rightarrow n! < \Big(\frac{n}{2}\Big)^{\!n} \enspace,$$
which is defined for all natural numbers.