I'm leaning towards the side of "functions", with some linguistic/philosophical caveats related to clearing up what exactly you and your friend were discussing/interested in knowing.
Operators act like functions...
I claim that anything you can do with an operator, you can do with a function, so that it comes down to whether an definition based on how to compute with them suffices*. If your friend agrees with me but says something "sure, I grant that every operator has an associated function that can do the same things, but they're not operators themselves", then I can't take sides without hearing their definition of "operator".
*(This could be considered an operational definition if you think about what the operators do or a theoretical definition since it's not about the way people use the word "operator".)
To this end, I'll try and challenge the examples your friend gave, one-by-one:
Integral
There are a few different ways you could interpret the indefinite integral operator. In one interpretation, it could be a function from, say, "the set $C(I)$ of continuous functions from an open interval $I$ to $\mathbb R$" to "the power set of $C(I)$" (sometimes denoted $\wp\left(C(I)\right)$) that sends a function $f$ to the set of functions $F$ defined on the same interval $I$ such that $F'=f$.
If we'd like the integral to have domain and codomain both being just a set of functions like $C(I)$, then the standard "indefinite integral" is a bit of an odd case simply because we don't usually care exactly which function we get. Basically, for each value of $a$, $f\mapsto g(x)={\displaystyle \int_a^x}f(t)\,\mathrm dt$ is an integral operator that's good enough, and as a sort of abuse of notation we bundle up the options together under the umbrella of the indefinite integral.
Square Root
I don't think it's really common to call this thing an operator, but we can certainly have a function from $\mathbb R$ or $\mathbb C$ to the power set of the same that outputs the set of all possible "square roots". I.e. given by $s(x)=\{y\mid y^2=x\}$. In this case, $s(4)=\{2,-2\}$, etc.
Less-than operator
I've definitely never heard of this operator, but we could still certainly have functions like $T:\mathbb R\to\wp(\mathbb R)$ given by $T(x)=\{y\mid y<x\}$.
There may be some confusion with the fact that in computer programming, we might refer to a different "less-than operator", whose function meaning is usually given by something like $L(x,y)=\begin{cases}\text{True}&\text{ if }x<y\\\text{False}&\text{ otherwise}\end{cases}$
Random-Number Generator
This is subtle, because I think the vast majority of mathematicians would not call that an operator, but in some programming contexts, the word "operator" pops up in related code (example). "Functions" in imperative programming languages aren't really like functions in math, but I would still say that a pseudorandom number generator at some level acts like a function (with suitably computerized domain and codomain) $U:\mathbb Z\to \mathbb [0,1]\times \mathbb Z$ where $U$ takes a current seed/state as input, and outputs a random number and a new seed/state.
...But there are two words for a reason
Operators can have different syntax
In contexts where "functions" need parentheses around their arguments, operators don't (e.g. $Df$ for the derivative of $f$). Or in the case of (partial) indefinite integrals, arguments get special notation: $\int f(x,y)\,\mathrm dx$ has the "variable of integration" argument after a special symbol $\mathrm d$. Or in the case of definite integrals or summation, the arguments get special placements: we write ${\displaystyle \sum_{i=1}^5} i^2$ instead of something like $\sum(i,1,5,i^2)$.
So you could say "some operators aren't functions because you can't write functions in weird ways like that". This is very similar to the question "Are infinite sequences just functions with domain the positive integers or not?". If the notation is part of what's under discussion, then they're not functions.
"Operator" is used in special contexts
Even restricting ourselves to cases/books where operators and other functions may look the same in symbols, just like there aren't really absolute synonyms, usage of the word "operator" isn't interchangeable with that of "function".
There are a variety of special implications/uses/contexts, and this won't be a comprehensive list, but some are:
- An operator is a linear map/function/transformation where the domain and codomain are the same vector space. (Sources: MIT OCW Quantum Physics II notes, Rudin)
- This is rarer, but some authors might generalize the above and speak of "linear operators" between two different spaces, or even call affine transformations "operators", as in page 7 of "Application of Distributions to the Theory of Elementary Particles in Quantum Mechanics", this MSE question, and these recent slides on "Affine computation and affine automation"
- An operator is something (often linear) that takes common functions (often $\mathbb R^n\to\mathbb R^m$ or $\mathbb C^n\to\mathbb C^m$) to other common functions, like differential operators (including things like the gradient), pseudo-differential operators, Schwarzian derivative (which is not linear, so this isn't a case of the first definition), integral transforms, and differintegral operators. In a context like this, "function" would be restricted to the things the operators can take as inputs, and the operators wouldn't be called "functions".
- As mentioned before, "operator" is used in programming to mean something very much like "a (built-in) function but with a special syntax".
So something called an "operator" in mathematics is essentially always something analogous to a linear transformation from a vector space to itself, with things like the Schwarzian derivative and a couple of exceptional generalizations pushing on that boundary a bit.
What's the verdict?
It depends what sort of meaning of "operators are functions" you both had in mind. In terms of inputs and outputs, they are all still functions. But in terms of usage and notation, they can act differently than arbitrary functions, and in many contexts would not be called "functions".
Best Answer
All vectors are sequences False To a mathematician, a vector is an element of a vector space, and therefore need not be a sequence. For example, the set of all continuous real valued functions on $[0,1]$ forms a vector space, and its elements are the 'vectors' of that space. Since the most familiar example of a vector space is $\mathbb R^n$, we often use the word 'vector' to refer to any (normally finite) sequence of elements, even when the elements do not naturally form a vector space; but it is certainly incorrect to say that all vectors form sequences.
All sequences are vectors Also false, but not for the reasons you have given. If the terms of a sequence are elements of a field $F$, then that sequence is an element of the vector space, over $F$, of all sequences with terms in $F$, so it is a 'vector' in that sense. However, if the terms of the sequence are sets with no extra structure, points of a metric space etc., then it doesn't make sense to say that the sequence is a vector. Note that this has nothing to do with the sequence being infinite dimensional. There exist infinite-dimensional vector spaces; indeed, dimension as a concept doesn't enter into the definition of a vector space at all: to be a vector, we onlyneed to be able to add and subtract with other vectors, and to be multiplied by scalars. It happens that the simplest examples of vector spaces are finite-dimensional.
All sequences are functions True, at least in the sense that we can view a sequence as a function from $\mathbb N$ into some set $A$. Bear in mind, though, that this is but one formalism of the idea of a sequence; if we implemented the sequence in some other way - say, as the set of all its finite initial segments: $$ \{(a_1), (a_1,a_2), (a_1,a_2,a_3),\dots\} $$ or in any one of a number of possible ways that don't involve functions, it would still be a sequence; it just happens that the function definition is the most convenient.
All functions are sequences Certainly false, as you correctly demonstrated.
All functions are operators I'd say false, because I normally hear the word operator used to describe particular examples of functions: binary operators like $+,-,\times$ etc. or linear operators in functional analysis. Then again, I wouldn't feel it was wrong to declare that 'operator' and 'function' should be synonyms. However...
All operators are functions This is definitely true; it is certainly not the case that functions can only map numbers to numbers: in mathematics, a function can map from any set to any other set.
Sorry if some of that went over your head. The point I'm trying to make is that the concepts 'vector', 'sequence' and 'function' come from different areas in mathematics (though of course they are used together all the time), while 'function' and 'operator' are very similar words (with 'function' being the more fundamental and universal term in case of doubt). So there is no such hierarchy in mathematics. But well done for exploring and trying to find patterns; keep it up and you'll discover some really beautiful ones.