Exercises 1

1. Suppose that Achilles gives the tortoise 100 metres start and that he runs this distance (at a constant velocity) in 10 seconds. Suppose that he runs 17 times as fast as the tortoise. How soon does he catch up ?
   Write down the infinite (Geometric) series that one obtains by taking the time at which he gets to where the tortoise was the last time the measurement was made, and show (using your skills of summing a Geometric Progression) that this gives the same answer.

   Solution to question 1

2. Prove or disprove the following, where A, B, C and D are sets.
   (a) (C - A) ∩ (C - B) = C - (A ∪ B)
   (b) (A ∩ B ∩ C) - D = (A - D) ∪ (B ∩ C)
   (c) (A ∪ B) × (A ∪ C) = (A × A) ∪ (A × C) ∪ (B × A) ∪ (B × C)
   [To prove that two sets X and Y are equal, first show that if x ∈ X then x ∈ Y, so that X ⊆ Y. Then prove that Y ⊆ X. To show sets are not equal, find an element which is in one of them but not in the other. For identities involving three sets, you can use a Venn diagram.]

   Solution to question 2

3. Determine which of the following are true.
   (∀x ∈ Z)(∃y ∈ Z)(x/y > 0)
   (∃x ∈ Z)(∀y ∈ Z)(x/y > 0)
   (∀x ∈ Z)(∃y ∈ Z)(∀z ∈ Z)(x + y > z ⇒ x - y > z)
   If P is a statement prove that the statement ¬(∀x)P is equivalent to (∃x)(¬P)

   Solution to question 3

4. Let r ≠ 0 be a rational number and let x be irrational. Show that x + r, x - r and x × r are all irrational.
   If x and y are irrational, are x + y, x - y, x × y and x / y necessarily irrational ?
   If r and s are rational numbers with r < s, show that there is an irrational number x satisfying r < x < s.

   Solution to question 4

5. If an integer has a rational square root, show that the square root must be an integer.
   For which integers n is √(n + 4) + √(n - 4) rational ?
   If b is an integer, prove that the quadratic equation x²+ 2bx + 2 = 0 never has rational solutions.

   Solution to question 5

6. Adapt the method used in lectures to prove that √2 is irrational to show that √3 is irrational.
   Prove that √6 is irrational and hence or otherwise prove that √2 + √3 is irrational.
   (Harder) If m and n are integers with irrational square roots, prove that √m + √n is irrational.

   Solution to question 6

7. Let P be the statement: The statement on the other side of this card is True and let Q be: The statement on the other side of this card is False.
   Write P on one side of a card and Q on the other and show that the system you get is inconsistent.
   What about the systems you get if you write P on both sides of the card or Q on both sides of the card ?

   Solution to question 7