In the context of set theory, I see on websites that set builder notation like $$\{x \mid P(x)\}$$ is read in natural language as "the set of all $x$'s that satisfy the predicate $P(x)$". But since a variable ($x$ in this example) can hold any value but not multiple values, what does "the set of all $x$'s" mean?
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$\begingroup$ The symbol "$x$" represents one value. When we say $\{x: P(x)\}$ we mean every individual value that has the property defined by $P$. That's why it's called a variable and not a constant. For example, let $P(x)$ be the statement that $x$ is an even number. Then $P(2), P(4), P(6), .. \text{ etc}$ are all true statements so those values, $2,4,\dots$ are in the set. $\endgroup$– John DoumaCommented Jun 26 at 16:23
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$\begingroup$ Could you expand in an answer, thanks for your effort $\endgroup$– John gregCommented Jun 26 at 16:27
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2 Answers
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Actually a more precise notation would be something like $\{x \in S \mid P(x)\}$, i.e. "the set of all members $x$ of the set $S$ such that $P(x)$". Think of looking through all the members of the set $S$: if a member satisfies the predicate $P$, you put it in your set, otherwise you don't. We often leave out the "$\in S$" if it's clear from the context what $S$ should be. But there is no such thing as "the set of all $x$" if you don't have such a context.
answered Jun 26 at 16:43
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$\begingroup$ So what does "all members $x$ mean"? How can many things(members of a set ) be one thing($x$) $\endgroup$ Commented Jun 26 at 16:51
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4$\begingroup$ What's wrong with "The set of all students who have a GPA at least $3$"? "Students" is plural but you are thinking about them one at a time. $\endgroup$ Commented Jun 26 at 17:05
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$\begingroup$ @EthanBolker The problem is it’s inconsistent: in your example “x” would be “students”, but P(x) would be the proposition “x has a GPA at least 3”, implying that “x” refers to one specific student. How could the same x refer to different things? $\endgroup$ Commented Jun 27 at 11:51
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$\begingroup$ @Divide1918: I don't find that observation that compelling. One could just as easily write "the set containing each $x$ such that $P(x)$." I don't think the vagaries of English grammar reflect strongly on that notation. $\endgroup$ Commented Jun 28 at 5:10
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$\begingroup$ @BrianTung I decided to incorporate your suggestion into my answer below. $\endgroup$– ryangCommented Jun 30 at 15:49
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In this discussion, element means element of the domain of discourse.
$$\{x \mid P(x)\}$$ is read in natural language as
- the set of all $x$'s that satisfy the predicate $P(x)$.
This reading misrepresents the dummy variable $x$ $\big(\{x:P(x)\}=\{y:P(y)\}\big)$ as a free variable with a mystery referent. Furthermore, since the set's representative element $x$ stands for a proper noun (e.g., $7,$ Tony) rather than a common noun (e.g., integer, boy), the phrase “all $x$'s” is quite loose. Clearer:
the set of all elements that satisfy $P$
$\{x\mid x<10\}$ is the set of all elements that are smaller than $10$
the set of all elements that each/individually satisfy $P$
$\{x\mid \exists n{\in}\mathbb Z\: (x=\frac n3\pi\,\text{ or }\,x=\frac{2n+1}5\pi)\}$ is the set of all elements that are each, for some integer $n,$ equal to $\frac n3\pi$ or $\frac{2n+1}5\pi$ (The choice of $n$ varies with the elements and isn't fixed for the set as a whole.)
$\{x\mid x^2<9\,\text{ or }\,x>20\}$ is the set of all elements that each either have a square smaller than $9$ or are bigger than $20$ (The Either Or applies individually to the elements rather than to the set as a whole.)
the set containing every element $x$ such that $P(x)$
the set of all elements, each denoted by $x,$ such that $P(x)$
the set of all elements $x$ such that $P(x)$.
And $\,\{f(t) \mid P(t)\}\;$ is the set of all elements of the form $f(t)$ such that $P(t)$.
Why would we denote every element by $x$? Doesn't that mean that $x$ holds many values?
In the notation $S=\{\color\red t\mid \color\red t \text{ is even}\}=\{\ldots,-4,-2,0,2,4,6\ldots\},$ each instance of the argument $t$ of the predicate $P(t)$ refers to a single element, rather than every element all at once. The placeholder $\color\red t$'s function is just to connect the left and right sides of the ‘such that’ symbol ‘∣’, drawing elements from the background set one at a time, in each instance checking whether that element satisfies $P(t),$ and collecting into $S$ precisely those that pass this test (e.g., accepting ‘8’ and rejecting ‘9’).
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$\begingroup$ Why would we denote every element by $x$ doesn't that mean that $x$ holds many values $\endgroup$ Commented Jun 26 at 17:43
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$\begingroup$ Sorry to ask this because it seems stupid, but what is an instance of a variable,what does instance of $x$ mean? $\endgroup$ Commented Jun 27 at 9:42
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$\begingroup$ @Johngreg Different occurrences/instances of $t$ have different values because $t$ is variously representing different elements. In the statement "every $t$ is greater than 0 $(\forall t\;t>0)",$ where the universe is the set of positive integers, each instance of $t$ is pointing at a different positive integer:
1is bigger than 0 and2is bigger than 0 and3is bigger than 0 and so on. The instances of $t$ within $\mathbb C$ that satisfy $t^2=1$ have values-1and1; when the function $t^2$ is graphed, each of the infinitely many instances of $t$ is represented by a vertical line. $\endgroup$– ryangCommented Jun 27 at 11:40 -
$\begingroup$ Unfortunately in natural language we do not have variables, so we tend to resort to circumlocutions. Thus you might read $\{x \in \mathbb R \mid x^2 < 10 \}$ as "the set of all real numbers whose square is less than $10$." That's not so bad, but it becomes difficult when the condition is more complicated. $\endgroup$ Commented Jun 28 at 19:42
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$\begingroup$ @RobertIsrael Eh? No circumlocution (using unnecessarily many words to express an idea especially in an evasive/indirect/vague way) here. $\endgroup$– ryangCommented Jun 30 at 14:38