1. Consider the following statement: "For every natural

number n, there exists a prime number p such that p > n". Is

this statement true or false? Explain your reasoning.

 - A) True, because there are infinitely many prime

numbers.

 - B) False, because there are only finitely many prime

numbers.

 - C) True, because of Euclid's theorem.

 - D) False, because of the prime number theorem.

 - Answer: C) True, because of Euclid's theorem. Euclid's

theorem states that for any natural number n, there is a

prime number p greater than n. This can be proved by

considering the product of the first n primes plus one,

which is not divisible by any of the first n primes, and

therefore must have a prime factor greater than n.

2. Let A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. What is the

cardinality of A ∪ B? Show your work.

 - A) 4

 - B) 5

 - C) 6

 - D) 8

 - Answer: C) 6. The cardinality of a set is the number of

elements in the set. To find the cardinality of A ∪ B, we

need to count the number of elements that are in either A or

B or both. We can use the formula |A ∪ B| = |A| + |B| - |A

∩ B|, where |A ∩ B| is the cardinality of the intersection of

A and B. In this case, |A| = 4, |B| = 4, and |A ∩ B| = 2,

since A and B have two elements in common: 2 and 4.

Therefore, |A ∪ B| = 4 + 4 - 2 = 6.