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I'm reading Coding the Matrix and in section 0.3.3 Klein says:

"For sets D and F, we use the notation FD to denote all functions from D to F. For example, the set of functions from the set W of words to the set ℝ of real numbers is denoted ℝW.
This notation derives from a mathematicla "pun":
Fact 0.3.9: For any finite sets D and F, |DF|=|D||F|."

I don't get the pun. Please explain.

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    It's not a pun. The fact is that if $D$ has $x$ elements and $F$ has $y$ elements, there are $y^x$ functions from $D$ to $F$. Using common notation for cardinality, the notation $F^D$ for the set of functions "agrees" with a formula for cardinalities. So you only need to "remember" one thing instead of two. Still not a pun though; they need to read again the dictionary definition of "pun". (Speaking of which - the formula is incorrect when $x=y=0$ because $0^0$ is undefined, but some people - including all computer people - insist to define $0^0=1$ exactly for this reason.) – 2017-01-20
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    I thought puns were funny – 2017-01-20
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    @user4757074 Good one. – 2017-01-20
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    @user4757074 They are. This one's not funny so by contrapositive it's not a pun. Notating the set of functions that way is usually called "suggestive notation," or notation by analogy with the case of finite cardinalities. Certainly not a pun IMO. – 2017-01-20
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    @mathguy: it's just your opinion that $0^0$ is not undefined: it has an agreed and precise definition by the many mathematicians, computer scientists and others who disagree with your value judgment about it. (They find your idea that the product of an empty subset of a ring should not be its unit element to be bizarre.) – 2017-01-20
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    @RobArthan - That is certainly not "just my opinion." Like I said, there is a very large group of people who choose to define it that way - in a way they would never be able to define either $0.5^2$ or $2^{0.5}$. – 2017-01-20
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    @mathguy: sorry. I should have written "it's a widespread opinion that $0^0$ should be considered to be undefined". (I see that we made complementary typos: there is a "not" in my comment that actually belongs in yours $\ddot{\smile}$.) – 2017-01-20
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    The notion of a "pun", in this sense, is not one that has anything to do with humor. It has to do more with the use of terminology based on a superficial analogy between distinct notions. Another example is the idea of an "adjoint functor" in category theory, which borrowed its name from adjoint operators in functional analysis on the basis of a similarity in notations. – 2017-01-20
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    @MaliceVidrine: what's superficial about these analogies? – 2017-01-20
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    @RobArthan - In the example I gave, neither version of "adjoint" is an instance of the other in any meaningful way. In the exponent notation for function spaces, the analogy comes from the fact that there's a nice numerical relation between natural numbers and finite cardinals, and between exponents of non-negative integers and the cardinal numbers of functions between finite sets; in the infinite case, pointing out that $|A^B|=|A|^{|B|}$ is question-begging because in the first instance there's no obvious "numerical" meaning of the right half of the equation. The pun comes first, here. – 2017-01-20
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    @MaliceVidrine: I don't understand (1) why you think analogies that aren't easily explained are superficial, (2) what your remarks about "question-begging" relate to in the present discussion and (3) what you mean by your valedictory remark that "The pun comes first, here". – 2017-01-20
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    @RobArthan - I think this discussion spills beyond the bounds of a comments thread, but to answer only (1), this isn't a matter of "easily explained"; adjoint operators and adjoint functors are simply entirely different things. – 2017-01-20

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The "pun" is that $F^D$ denotes a set, with no actual notion of "powers" as in two-to-the-tenth-power is 1024. But it happens that when you compute the cardinality of that set, it's actually $|F|^{|D|}$, which is something that does have to do with powers. The the "pun" is in the two different-but-related uses of the single notation "something-superscript-something-else".

If you look at Halmos's book on Naive Set Theory, you can see this "pun" extended further, in the sense that the number "2" in that book is defined to be the set containing $0$ and $1$, so $|2^F| = 2^{|F|}$, where the two $2$s represent slightly different things.

Note: from one dictionary: "pun: a joke exploiting the different possible meanings of a word or the fact that there are words that sound alike but have different meanings." In this case the "word" is a notation, and the notations look alike rather than sounding alike. So maybe it's a meta-pun.

Not all puns are funny; much of John Donne's poetry is a testament to this.

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    Halmos was not punny about that, however: I would bet he considered something very natural and unalarming. – 2017-01-20
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    I didn't mean to suggest Halmos said anything about puns -- merely that if you follow his "definition" of $2$, then one of the "cardinality" symbols goes away. (And I know it's not "his" definition -- it's just one of the most accessible books that i know of to introduce it.) – 2017-01-20
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The other answers here are correct to the question I asked; but I believe it worth stating |D||F| is the same as permutations with replacement of F choose D which I find an interesting insight into generating the unique Domain - Co-domain pairs.