Vrije Universiteit Amsterdam
Wednesday 30 September, 2020
Basic concepts in mathematics, test ] 2.
You have 45 minutes to complete this test.
This test counts for 25% towards your final grade.
Name:
Question 1 (10%) Consider the statement
∼ (∀M < 0, ∃N > 0, x > N ⇒ f (x) < M )
Write down an equivalent statement by filling out the dots with the correct symbols. Choose symbols from
the list ∀, ∃, ⇒, ∧, ∨, <, >, ≤, ≥.
. . . . . . M . . . . . . 0, . . . . . . N . . . . . . 0, x . . . . . . N . . . . . . f (x) . . . . . . M
Question 2 Consider a collection of seven math books, four physics books and three chemistry books. All
books are different.
a) [10%] How many selections of six books contain exactly two math books? (An explanation is not
necessary – a number suffices.)
b) [10%] The books are placed on a book shelf. How many ways are there to arrange the books if books
of the same topic must be placed together? (An explanation is not necessary – a number suffices.)
Question 3 (20%) Let a ∈ Z and suppose that a17 + 17 is odd. Prove that a is even.
Question 4 (30%) Prove that
{n ∈ Z : 34 | n} = {n ∈ Z : 2 | n} ∩ {n ∈ Z : 17 | n} .
Question 5 (20%) Let p ∈ N be a prime number and let 1 ≤ a ≤ p − 1 be an integer. Prove that there
exists a b ∈ Z with
ab ≡ 1 (mod p) .
Hint: What is the greatest common divisor gcd(a, p)?