Python Set Comprehension

primes = {x for x in range(2, 101) if all(x%y for y in range(2, min(x, 11)))}

I simplified the test a bit - if all(x%y instead of if not any(not x%y

I also limited y's range; there is no point in testing for divisors > sqrt(x). So max(x) == 100 implies max(y) == 10. For x <= 10, y must also be < x.

pairs = {(x, x+2) for x in primes if x+2 in primes}

Instead of generating pairs of primes and testing them, get one and see if the corresponding higher prime exists.

You can get clean and clear solutions by building the appropriate predicates as helper functions. In other words, use the Python set-builder notation the same way you would write the answer with regular mathematics set-notation.

The whole idea behind set comprehensions is to let us write and reason in code the same way we do mathematics by hand.

With an appropriate predicate in hand, problem 1 simplifies to:

 low_primes = {x for x in range(1, 100) if is_prime(x)}

And problem 2 simplifies to:

 low_prime_pairs = {(x, x+2) for x in range(1,100,2) if is_prime(x) and is_prime(x+2)}

Note how this code is a direct translation of the problem specification, "A Prime Pair is a pair of consecutive odd numbers that are both prime."

P.S. I'm trying to give you the correct problem solving technique without actually giving away the answer to the homework problem.

You can generate pairs like this:

{(x, x + 2) for x in r if x + 2 in r}

Then all that is left to do is to get a condition to make them prime, which you have already done in the first example.

A different way of doing it: (Although slower for large sets of primes)

{(x, y) for x in r for y in r if x + 2 == y}