Write down the following verbal statements in formal notation.
Example. All real numbers are also rational numbers.
(i) The set contains the number .
(ii) The set contains the number but not the number .
(iii) No natural number is strictly negative (that is, strictly smaller than zero).
(iv) If is positive and is negative, then (this implies that) the product is negative.
(v) At any multiple of , the -function is equal to zero.
(vi) For any natural number and any integer, if the integer is positive, then their product is positive.
Write down the negation of the following verbal statements in formal notation, i.e. the formal statement that asserts the exact opposite of the verbal statement given (do not simply use in front of the statement!).
(i) is contained in the natural numbers.
(ii) At any multiple of , the -function is equal to zero.
(iii) The set contains some objects which are not real numbers.
(i) Suppose that and are statements that each assert some fact. What must we necessarily show to disprove “ and are true”, i.e. the statement ?
(ii) Consider again the setup of (i). Which points are sufficient do disprove ?
(i) Consider five statements . Suppose that , and . What can you say about the statements?
(ii) Consider again the setup of (i). Can you find a sufficient condition for ? What about and ?
Assess whether the following statements are necessary, sufficient, equivalent, or neither of the previous, for , where is a non-empty set.
This exercise is intended to train your understanding of how sets are helpful in thinking about statements and arguments. The solutions feature no drawings using the “circle approach” to sets, but you may find it helpful to draw the sets when assessing whether given relationships hold or not.
(i) Re-write the following statements and arguments in set notation:
Example: For 1., define as the set of unicorns, as Charlie and as the set of beings/objects that can be assigned the property of “liking candy”. Then, we can write 1. as . Using the subset notation, we can also find a more elegant way of writing this argument: .
(ii) Consider the arguments (1., 3., 5.) in (i). Are they valid?
Write down the formal notation for the sets verbally described in the following.
Hint: the “direct” explicit representation of the set may not always be the most efficient one.
(i) Compute union, intersection and the set differences for and .
For the questions to follow, recall that it may be helpful to illustrate sets using circles.
(ii) If , how do and relate to each other?
(iii) Consider the relationship . Is it true for arbitrary sets , and ? Can you give a reasoning for why or why not, respectively? What about the analogous statement when we change intersections to unions and vice versa?
Take the derivatives of the functions mapping from and to real numbers, with the following mapping rules:
(iii) (assume that we do not use arguments so that is always real)
Recall that for continuous functions , , i.e. we can pull the limit into continuous functions. In the following, you can take for granted that continuity applies to the exponential function, the cosine and sine function, the square root function and, when is a constant, polynomial functions and the “power” function .
Determine the limits of
(iii) (Hint: L’Hôpital’s rule)
This concludes the exercises for Chapter 0. If you desire to practice more, you may consult the recap questions in the companion script.