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I have 26 tiles each one with a different letter of the English alphabet: $A, B, C,...,Z.$

I draw two tiles with replacement. The possible outcomes are: $$S=\{AA, AB, AC,..., AZ, BA,...,ZZ\},$$ which contains $26^2=626$ elements.

Now define the random variable $X$ which counts the number of vowels I draw. X can take on 3 values: 0, 1, or 2. Let U={0,1,2} which is the set of possible values that X can take. I am describing is an elementary example of the Binomial distribution.

What I need help with is knowing the correct name for the set S, and the set U described above.

Is it outcome space, sample space, event space?

I know outcomes are things that can happen in an experiment like rolling a six-sided die the outcomes are S={1,2,3,4,5,6} but an event is some subset of the "set of all sets". For example the event might be rolling a prime number for which there are three possible values U={2, 3, 5}

What is S in both examples called? What is U?

Thanks.

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Sampe space refer to to $\Omega$ which is the set of all possible elementary events $\omega$. I don't really know what you mean by outcome space. If you mean the space of all possible outcomes, then you are referring to the sample space $\Omega$. If you mean the space, or rather set, of an outcome, you are referring to an event.

In your first example $U$ is the set which contains all possbible elementary events $\omega_i$, as such it is your sample space. The set $A=\{0,2\}$ could be one event, $B=\{2\}$ is another. When defining your new random variable $X$ you do not need to consider the previously defined set $S$.

In your second example, $S$ is your sample space and it contains all elementary, and $U$ is an event.

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