Lists and arrays, part by part
Jelly, 12 bytes

Task 1

r


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Task 2

r+þ0r$}


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Task 3

r+þ0r$}
×þ


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Task 4

r+þ0r$}
×þFṢ


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How it works

Task 1

r is the dyadic range atom and does exactly what the task asks for.

Task 2

A dyadic chain that begins with three dyadic links is a fork; the outmost links get evaluated first, then the middle link is called with the results to both sides as arguments.


r behaves as before, yielding [n, …, m].
0r$} is a quicklink (or quickquicklink, if you will).

The quick $ (monadich chain) consumes the links 0 (yield 0) and r (dyadic range) and turns them into a monadic chain. When called with argument k, this will yield [0, …, k].

The quick } (right argument) takes the quicklink created by $ and turns it into a dyadic link that calls 0r$ with 0r$}'s right argument.

0r$} will be called with left argument n and right argument m, so 0r$ is alled with argument m and yields [0, …, m].
+þ is another quicklink. þ (table) will call + (addition) for every element in its left argument and any element in its right argument, grouping the results for each right argument into a single row.

+þ will be called with left argument [n, …, m] and right argument [0, …, m], yielding the desired table.


Task 3

Every line in a Jelly program defines a different link. The last one is the main link and, like C's main function, is the only link that is executed by default. The remaining links can be called from the main link, but we won't do this here.

As before, þ (table) will call × (addition) for every element in its left argument and any element in its right argument, grouping the results for each right argument into a single row.

Since both arguments to ×þ are integers, þ casts them to ranges, transforming the arguments n and m into [1, …, n] and [1, …, m].

Task 4

×þ works as before. The following links are monadic, making them atops, i.e., they are applied on top of the previous ones.

After executing ×þ, F flattens the resulting 2D array, and Ṣ sorts the resulting 1D array.