What distinguishes 'family' vs. 'set' of functions?

Question

Source: Stewart, James. Calculus: Early Transcendentals (6 edn 2007).

[p. 50 Top:]   To understand how the expression for a function relates to its graph, it’s helpful to graph a family of functions, that is, a collection of functions whose equations are related. In the next example we graph members of a family of cubic polynomials.

[p. 391 Middle:]   You should distinguish carefully between definite and indefinite integrals. A definite integral $\int^b_a f(x) \,dx$ is a number, whereas an indefinite integral $\int f(x) \,dx$ is a function (or family [format mine] of functions).

1. Of functions: how does 'family' differ from 'set'?

2. Why did James Stewart write 'family' instead of 'set'?

I read this that feels too advanced for univariate calculus.

Answer 1

The expression a set of functions means a collection of functions, grouped together by some basic common property. For example, $$ L^2[0,1] = \left\{ f:[0,1] \to \mathbb{R} \left| \int_0^1 f(x)^2 dx < \infty\right.\right\}. $$ The relationship feature is common for all functions in the class, but not strong enough to call it a family.

A family of functions typically suggest a very strong relationship, in structure, or in form, perhaps differing by values of a couple of parameters. For example, a set of solutions to $y'(t) = y(t)$ is a family of functions $$y(t) = Ce^t,$$ a much stronger relationship than the one above.

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Your example from Stewart is a classic form of this usage. The expression $\int f(x) dx$ represents a family of functions $\mathcal{F} = \{F(x) + c\}$, different by the constant factor $c$ only, having the property that if $\phi \in \mathcal{F}$ then $\phi'(x) = f(x)$.

Answer 2

Here's a definition that might work (though it looks like the author's definition is more general and much fuzzier).

A family is a set that is related in a known way by parameters. Let's look at some examples.

Consider the equation for a line: $f(x)=mx+b$; here $m$ is the slope and $b$ is the "y-intercept". Consider the case where $m$ is $1$ and $b$ is 0. If we graph it, we get a line at 45 degrees. What if $m=0$ and $b=0$? Then we get a horizontal line (that lies on the x axis). So you see that we get different lines by changing the slope and y-intercept. We can call these lines a family of lines; they're related by the parameters $m$ and $b$.

How about an indefinite integral? Suppose $f$ is the indefinite integral of $f'$. Then one indefinite integral of $f'$ is $f+1$. And another indefinite integral is $f+2$. And yet another is $f+0$. So you see that a general expression of the families of indefinite integrals is $f+c$. And that's the family of functions related by the parameter $c$.

Answer 3

Given a ground set $X$ of "animals" (objects of type "animal") a set of animals is a subset $A$ of $X$. The elements of $A$ are neither ordered, nor numbered, nor is there any other "organizing" structure implied on $A$. Any "animal" $x\in X$ is either an element of $A$, or not. Basta.

Now families. Given an arbitrary "index set" $I$, a function $$f:\quad I\mapsto X\qquad \iota\mapsto x_\iota$$ names for each $\iota\in I$ a certain animal $x_\iota \in X$, whereby the same animal can be named several times. If we are not interested in the abstract triple $(I,X,f)$ per se, nor on some extraneous properties of $f$, like continuity, etc., but only on the list (or array) of animals produced (or organized) by $I$ and $f$, then we denote this list by $(x_\iota)_{\iota\in I}$, and call it a family of animals.

A definite integral $\int_a^b f(x)\>dx$ is a number. An indefinite integral $\int f(x)\>dx$ is a set of functions, namely the set of all functions $F$ satisfying $F'=f$ (on some agreed interval). But $\int f(x)\>dx$ is not a family of functions, whereas $\bigl(x^3+C\bigr)_{C\in{\mathbb R}}$ is. Think about it!

Answer 4

This is a definition of family in "Dictionnaire des Mathématiques Modernes" of Larousse.

Let $S$ and $I$ be two sets. An injection of $I$ into $S$ is called a family of elements of $S$ indexed by $I$ and it is noted $i\to x_i$ or $\{x_i\}_ {i \in I}$. The set $I$ is called then set of indices.

Every subset $P$ of $S$ can be considered as a family of elements of S, thanks to the canonical injection of $P$ in $S$.