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Can someone explain to me what function spaces and what functon ev are because I couldn't understand the concepts from my lecture slides? And please help me answer the questions below. Thanks.

Function spaces

Given sets $X$ and $Y$, let $Y^X$ denote the set of all functions from $X$ to $Y$

• if $X$ and $Y$ are finite sets, what is the cardinality of $Y^X$ ?

• There is an important function $\textsf{ev}: X\times Y^X \rightarrow Y$. Can you guess how it is defined?

Question 1:

Write down all the elements of the following function spaces: $2^0$, $2^1$ and $2^2$

Question 2:

A picture of the second question

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    @Legoman If you don't understand the concept, what do you suggest one should try? – 2017-01-19
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    I just don't understand the concepts. That's why I haven't made any attempts to answer the questions. – 2017-01-19
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    @skyking I know the concept, I would know what he tried to do – 2017-01-19
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    @user408340 Ok, I thought you had some problems to solve it. – 2017-01-19
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    @Legoman What he tried to do is to get somebody that know the concepts to explain them to him. I think that a reasonable thing to try. Otherwise it would not help that other people know the concepts... – 2017-01-19

1 Answers 1

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For function space one need to realize that a function is a mathematical object like any else and you can have a set of functions. Such a set is called function space.

The second point is about what the function $ev$ is. It's first asked as a question if you can guess what it is, but then in question two it is defined so you can either read there or get the explanation here:

$ev$ seem to mean "evaluate". It's a function that takes a function and a value and applies the function to the value. For example if you have $A=\{a\}$, $B=\{b\}$ and a function $f: A\to B$ (which obviously must map $a$ to $b$) you would have $ev(f, a)=b$

For example if you have a function $i$ that maps $a$ to take the function $\cos$ and apply to $0$ you get $1$ so $ev(\cos, 0) = 1$.

For the question one I assume that you would need to know what the sets $0$, $1$ and $2$ denotes. Perhaps it's not standard notation, but I'd guess that they are arbitrary sets of that size, ie $0 = \emptyset$, $1 = \{a\}$ and $2=\{a, b\}$ (for some $a\ne b$).

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    Can you explain what you mean by "cos"? – 2017-01-19
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    It's the cosine function. If you have not learnt about it, it's sufficient to know that $\cos 0=1$. – 2017-01-19
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    Anyway I rephrased that to skip references to the $\cos$ function. – 2017-01-19
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    So 0, 1 and 2 in the first question are cardinalities, right? – 2017-01-19
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    And is ev(f,a) = b the same as f(a) = b? – 2017-01-19
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    @user408340 I think so about $0$, $1$ and $2$, but I'm not sure that this is standard notation (you will have to check with your literature). And yes $ev(f,a)=b$ is the same as $f(a) = b$. – 2017-01-20