Set-builder notation is an alternative method of writing down sets. In the current context, the mathematical punctuation mark colon “∶” will mean “such that”.
Example 35. The set {x ∈ R ∶ x > 0} contains all x ∈ R such that x > 0. In words, this set contains all real numbers that are positive.
What comes after the colon are the conditions or criteria that x must satisfy, in order to qualify as a member of the set. Our sets will usually contain only numbers, but here’s an example to show you how we can write down one particular set of musical artists.
Example 36. The set {x ∶ x is an artist that has had a US Billboard Hot 100 #1 Single}
contains all the artists who have ever had a US Billboard Hot 100 #1 Single.
It will however be more typical for our sets to be sets such as these:
Example 37. {x ∈ R ∶ x > 0} = R+, Q+ = {x ∈ Q ∶ x > 0}, Z+ = {x ∈ Z ∶ x > 0}, R+0 = {x ∈ R ∶ x ≥ 0}, Q+0 = {x ∈ Q ∶ x ≥ 0}, and Z+0 = {x ∈ Z ∶ x ≥ 0}.
Remark 2. We use the colon ∶ but some writers use instead the pipe ∣.
Exercise 24. Write down R−, Q−, Z−, R−0, Q−0, and Z−0 in set-builder notation. (Answer on p. 1009.)
Exercise 25. Write down (a, b), [a, b], (a, b], and [a, b) in set-builder notation. (Answer on p. 1009.)
Exercise 26. Let X = {x ∶ x is a living current or former Prime Minister of Singapore}.
Write down the set X so that all its elements are explicitly stated. (Answer on p. 1009.) Exercise 27. Rewrite each of the following sets in set-builder notation: (a) (−∞, −3) ∪ (5, ∞). (b) (−∞,√
2] ∪ (e, π) ∪ (π, ∞). (c) (−∞, 3) ∩ (0, 7). (Answer on p. 1009.)
2 Dividing By Zero
This very brief chapter is to warn you against making a common mistake — dividing by 0. Students have little trouble avoiding this mistake if the divisor is obviously a big fat 0.
Instead, students usually make this mistake when the divisor is an unknown constant or variable that might be 0.
Example 38. Find the values of x for which x(x − 1) = (2x − 2)(x − 1).
Here’s the wrong solution: “Divide both sides by x− 1 to get x = 2x − 2. So x = 2.”
Here’s the correct solution: “Case #1. Suppose x− 1 = 0. Then the given equation is satisfied. So x = 1 is one possible value for which x(x − 1) = (2x − 1)(x − 1). Case #2.
Now suppose x− 1 ≠ 0. So we can divide both sides by x − 1 to get x = 2x − 2. So x = 2.
Conclusion. The two possible values of x for which x(x − 1) = (2x − 1)(x − 1) are x = 1 and x= 2.”
Moral of the story. Whenever you divide by a certain quantity, make sure it’s non-zero.
If you’re not sure whether it equals 0, then break up your analysis into two cases, as was done in the above example: Case #1 — the quantity equals 0 (and see what happens in this case); Case #2 — the quantity is non-zero (in which case you can go ahead and divide).
By the way, let’s take this opportunity to clear up another popular misconception — You may have heard that 1
0 = ∞. This is wrong. 1
0 ≠ ∞. Instead, any non-zero number divided by 0 is undefined.14 “Undefined” is the mathematician’s way of saying, “You haven’t told me what you are talking about. So what you are saying is meaningless.”15
Exercise 28. What’s wrong with this “proof” that 1= 0? (Answer on p. 1010.) 1. Let x, y be positive numbers such that x= y.
14Once again, the truth is actually somewhat more complicated. Under certain special contexts in more advanced mathe-matics, 1
0 is well-defined. But in this textbook, I’ll simply keep it simple and insist that 1
0 is undefined.
15On the other hand, 0
0 is indeterminate, which means that it’s typically undefined, but can sometimes be defined under certain circumstances.
3 Functions
Undoubtedly the most important concept in all of mathematics is that of a function — in almost every branch of modern mathematics functions turn out to be the central objects of investigation.
- Michael Spivak (1994 [2006], Calculus, p. 39).
You are probably familiar from secondary school with such statements as: “Let f(x) = x+8 be a function.” Strictly speaking, this is not the correct way of describing a function.
Here is a more precise definition of a function.16 A function consists of three pieces:
1. A set called the domain;
2. A set called the codomain; and
3. A mapping rule (or simply mapping or simply rule) which specifies how each and every element in the domain is mapped (or assigned) to one (and exactly one) element in the codomain.
Remark 3. The codomain is not the same thing as the range. We’ll learn about the range only in the next section.
Altogether then, a function simply maps (or assigns) each element in the domain to one (and exactly one) element in the codomain.
Example 39. Let f be the function whose ...
• Domain is the set {Cow, Chicken};
• Codomain is the set {Produces eggs, Produces milk, Guards the home}; and
• Mapping rule is, informally, “match the animal to its role”.
According to the mapping rule, “Cow” (in the domain) is mapped to “Produces milk”
(in the codomain) and “Chicken” (in the domain) is mapped to “Produces eggs” (in the codomain). Every element in the domain is mapped to exactly one element in the codomain.
This is indeed a function, because it has a domain, codomain, and a correctly-specified mapping rule.
Example 40. Let f be the function whose ...
• Domain is the set {1, 2};
• Codomain is the set {1, 2, 3, 4, 5}; and
• Mapping rule is, informally, “multiply by 2”.
According to the mapping rule, “1” (in the domain) is mapped to “2” (in the codomain) and “2” (in the domain) is mapped to “4” (in the codomain). Every element in the domain is mapped to exactly one element in the codomain.
This is indeed a function, because it has a domain, codomain, and a correctly-specified mapping rule.
Example 41. Let f be the function whose ...
• Domain is the set R;
• Codomain is the set R; and
• Mapping rule is, informally, “round off to the nearest integer, where half-integers are rounded up”
According to the mapping rule, “3” (in the domain) is mapped to “3” (in the codomain),
“3.14159” (in the domain) is mapped to “3” (in the codomain), “3.5” (in the domain) is mapped to “4” (in the codomain), and “3.88” (in the domain) is mapped to “4” (in the codomain). Every element in the domain is mapped to exactly one element in the codomain.
This is indeed a function, because it has a domain, codomain, and a correctly-specified mapping rule.