Writing sets: roster, rule and the empty set
How do you write down exactly which things belong to a set, when there are a few of them, a great many, endlessly many, or none at all?
Where you are 15 in Sets
Why this exists
A set is decided by what belongs to it (elements and membership). That settles what a set is, but not how to write one down, and the sets mathematics needs come in very different sizes.
Some are small with no pattern, like the people in a club, and the only way to say who is in them is to name names. Some are large but regular: there are 142 positive multiples of 7 below 1000, and nobody wants to write them all when the pattern is plain after three. Some never end, like the positive multiples of 5. Some have nothing in them, like the whole numbers between 7 and 8.
So mathematics uses a few ways of writing sets: a list for the small ones, a list with an ellipsis for the regular ones, a rule in words for any set at all, and a symbol for the set with nothing in it.
Definition
Roster notation. A set is written by listing its elements between braces, separated by commas. The list of entries is called a roster.
This is the set whose elements are exactly 11, 13, 15, 17 and 19, and nothing else. Order and repeats do not matter: \{13, 11, 19, 17, 15\} is the same set, and writing 13 twice adds nothing.
A rule in words. A set can also be described by a condition that every object either meets or fails. The set above is “the odd whole numbers between 10 and 20”. The rule and the roster name the same set, because they let in exactly the same objects. A rule must give a clear yes or no for every object it could be asked about; “the big numbers” is not a rule, because it gives no answer for 60.
The ellipsis. Three dots, \dots, read “and so on”, mean that the entries carry on in the pattern the first few have set up.
- With a last entry after the dots, the list stops there: \{5, 10, 15, \dots, 60\} is the multiples of 5 from 5 to 60.
- With nothing after the dots, the list never stops: \{5, 10, 15, \dots\} is all the positive multiples of 5.
- With the dots at the front, the list never stops going the other way: \{\dots, -15, -10, -5\} is all the negative multiples of 5.
The empty set. The set with no elements at all is the empty set. It is written either as braces with nothing between them, \{\ \}, or with the symbol \varnothing, read “the empty set”. The symbol is a circle with a slash through it; it is not the number 0.
Finite and infinite. A set is finite when its elements can be counted and the count comes to an end at some whole number. The empty set is finite, with 0 elements. A set that is not finite is infinite: however many of its elements have been counted, more are left.
Words for numbers in this lesson. The whole numbers are 0, 1, 2, 3 and so on, so 0 is a whole number. The integers are the whole numbers together with their negatives. Positive means greater than 0. A perfect square is the square of a whole number, so 0 = 0^2 is one.
Intuition
A roster is a guest list: here is everyone, by name. A rule is a door policy: anyone with a ticket may come in. A guest list suits a small party and is useless for a stadium; a door policy works for both. The ellipsis names a few guests and says “carry on the same way”, which works only when everyone would carry on the same way.
For the empty set, think of sets as boxes. The empty set is an empty box. The set \{0\} is a box holding one card that says 0: not empty, since it holds something. The set \{\varnothing\} is a box holding an empty box, and it is not empty either.
Results
A roster names exactly its entries. The set \{a, b, c\} has the listed objects as its elements and nothing else, in any order and however often each is written.
Every finite set can be written as a complete roster, and no infinite set can. An infinite set needs an ellipsis or a rule.
There is exactly one empty set. Any two descriptions that let nothing in, such as “the whole numbers between 7 and 8” and “the months with fewer than 28 days”, name the same set, \varnothing.
The sets \varnothing, \{0\} and \{\varnothing\} are three different sets, with 0, 1 and 1 elements.
An ellipsis stands for a rule. It names a set only when the pattern can be continued in just one way. When it cannot, the rule has to be written out in words.
Why it is true
Finite sets can be listed and infinite ones cannot. If a set is finite with n elements, writing each down once gives a complete roster of n entries. Going the other way, any list written out in full has some number of entries, say n, so the set it names has at most n elements and is finite. So no written list can name an infinite set.
There is only one empty set. Suppose E and F both have no elements. Two sets are equal when every element of each is an element of the other (subsets and equal sets). To break that we would need an element of E that is not in F, and E has no elements at all; the same holds the other way round. So E = F, which is why we speak of the empty set.
The three sets \varnothing, \{0\} and \{\varnothing\} differ. Each comparison needs one object that is in one set and not the other. Since 0 \in \{0\} but 0 \notin \varnothing, the first two differ. Since \varnothing \in \{\varnothing\} but nothing is in \varnothing, the first and third differ. And in this lesson a number is not a set, so 0 \notin \{\varnothing\}, and \{0\} \neq \{\varnothing\}.
An ellipsis stands for a rule. The dots are not an element; they say “continue the pattern”. Whether 40 belongs to \{5, 10, 15, \dots\} is settled by the rule the reader recovers from the first entries, “positive multiples of 5”. If two rules fit those entries, some object gets a yes under one and a no under the other, and the notation names no single set (Example 5).
Worked examples
Writing a roster from a description
Write each set as a roster and say how many elements it has.
(a) The odd whole numbers between 10 and 20.
(b) The months of the year that have 31 days.
(c) The digits used in writing the number 2026.
Solution(a) Neither 10 nor 20 is odd, so whether “between” includes the ends does not matter. Going up in steps of 2 from 11 gives 11, 13, 15, 17 and 19; the next, 21, is too large.
The set has 5 elements.
(b) April, June, September and November have 30 days, February has 28 or 29, and the other seven months have 31:
The set has 7 elements. Calendar order helps the reader, but any order names the same set.
(c) The number 2026 is written with the digits 2, 0, 2 and 6. A set records only whether something belongs, not how often, so the set is \{0, 2, 6\}, with 3 elements.
Reading an ellipsis
Let
Describe each set in words and say whether it is finite or infinite. Count the elements of the finite one. Then decide whether 45, 62 and 1000 belong to each set.
SolutionThe entries of A and of B start at 5 and go up by 5, so both are made of positive multiples of 5. The difference is what comes after the dots.
A stops at its last entry, 60: it is the multiples of 5 from 5 to 60. Its elements are 5 \cdot 1, 5 \cdot 2, \dots, 5 \cdot 12, so A is finite with 12 elements.
B has nothing after the dots: it is all the positive multiples of 5. It is infinite, because adding 5 to any element gives another element.
C has its dots at the front. Read backwards its entries go down by 5 for ever: C is the set of all negative multiples of 5, and it is infinite.
Now the three numbers.
- 45 = 5 \cdot 9 is a positive multiple of 5 between 5 and 60, so 45 \in A and 45 \in B. It is positive, so 45 \notin C.
- 62 = 5 \cdot 12 + 2 is not a multiple of 5 at all, so it belongs to none of the three.
- 1000 = 5 \cdot 200 is a positive multiple of 5, so 1000 \in B. It is past 60, so 1000 \notin A, and it is positive, so 1000 \notin C.
From a roster to a rule
Describe each set by a rule in words.
A rule describes a set correctly only if it lets in every element and nothing else, so each guess has to be tested both ways.
For P, every entry is prime and the largest is 29. Guess: “the prime numbers less than 30”. Test it the other way: 1 is not prime, and every whole number from 2 to 29 missing from the list, such as 9 = 3 \cdot 3 or 21 = 3 \cdot 7, has a factor other than 1 and itself. So P is the set of prime numbers less than 30.
For Q, the entries are 1^2, 2^2, \dots, 6^2, the squares of the whole numbers from 1 to 6. The rule “the perfect squares less than 40” fails the test: 0 = 0^2 is a perfect square less than 40, and 0 \notin Q. One more word repairs it: “the positive perfect squares less than 40” lets in exactly the six entries.
For R, one rule is “the integers from -2 to 2”; another is “the integers whose distance from 0 is at most 2”. The word “integers” matters: “the numbers from -2 to 2” would also let in \tfrac{1}{2}, which is not in R.
The empty set and its look-alikes
Write each set as a roster and say how many elements it has.
(a) The whole numbers between 7 and 8.
(b) The solutions of 3x = 0.
(c) The solutions of x + 4 = x + 9.
(d) The months with fewer than 28 days.
Solution(a) No whole number lies strictly between 7 and 8, which are next to each other. The set is \varnothing, with 0 elements.
(b) Dividing both sides of 3x = 0 by 3 gives x = 0, and 3 \cdot 0 = 0 checks, so the set is \{0\}, with 1 element. It is not empty: having zero as an element is not the same as having no elements.
(c) Subtracting x from both sides leaves 4 = 9, false whatever x is. No number is a solution, so the set is \varnothing.
(d) The shortest month, February, has 28 days, so no month has fewer. The set is \varnothing.
Sets (a), (c) and (d) are described in completely different ways, but each has no elements, so they are the same set, \varnothing. Only (b) differs. Figure 1 shows a third look-alike, \{\varnothing\}, which has one element, the empty set.
When an ellipsis is not enough
Someone writes S = \{1, 2, 4, \dots\}. Give two different rules that both begin 1, 2, 4, and find the next two entries under each. Then name a question about S that cannot be answered.
SolutionRule one: each entry is double the one before. That gives 1, 2, 4, 8, 16, \dots, the powers of 2. The next two entries are 8 and 16.
Rule two: add 1, then 2, then 3, adding one more each time. That gives 1, 2, 4, 7, 11, \dots, since 4 + 3 = 7 and 7 + 4 = 11. The next two entries are 7 and 11.
Both rules fit the three entries written. Under rule one 7 \notin S; under rule two 7 \in S. So “is 7 an element of S?” has no answer, and S as written is not yet a definite set. The cure is to say the rule in words: “S is the set of powers of 2: 1, 2, 4, 8, 16, \dots”. Showing more entries helps, but no number of entries rules out every other pattern; only the words settle it.
Boundaries
An ellipsis needs a pattern that can be continued only one way. Three entries that fit two rules, as in \{1, 2, 4, \dots\}, do not name a set. When there is any doubt, the rule goes in words beside the list.
An ellipsis needs a step from one element to the next. Between any two numbers lying between 0 and 1 there is another, so that set has no “next” member and the dots have no step to repeat. Sets like this are described by a rule, and the compact ways of writing such rules, set-builder notation and interval notation, are the subject of interval and set-builder notation.
A rule must give a definite yes or no, and say what kind of number it means. “The large multiples of 5” names no set, since nothing says whether 45 is large. “The numbers between 1 and 4” is \{2, 3\} if whole numbers are meant, and an infinite set if every number is meant.
Where this goes next
A rule in words can be written in symbols: set-builder notation writes “the numbers greater than 3” as \{x \mid x > 3\}, read “the set of all x such that x > 3”, and is taught with intervals in interval and set-builder notation.
The number of elements of a finite set is its cardinality, and sets that never run out are studied as infinite sets. Union and intersection build new sets from old ones, and the empty set turns up as the intersection of two sets with nothing in common. Pairing the elements of two sets gives the Cartesian product, and the domain and range of a relation are written as rosters (finding the domain and range).
Exercises
Writing a roster
- Write each set as a roster. (a) The even whole numbers between 31 and 41. (b) The integers strictly between -4 and 2. (c) The whole-number factors of 18. (d) The months whose names begin with the letter J.
- Write each set of solutions as a roster. (a) The solutions of x^2 = 64. (b) The solutions of (x + 1)^2 = 0.
- Which of these name the same set as \{10, 20, 30\}? (a) \{30, 20, 10\}, (b) \{10, 20, 20, 30\}, (c) \{10, 20, 30, 40\}, (d) the multiples of 10 between 5 and 35, (e) the whole numbers less than 35 that are multiples of 10 (remember that 0 is a whole number).
Ellipses, finite and infinite
- Say whether each set is finite or infinite, and give the number of elements of each finite one. (a) \{8, 16, 24, \dots, 80\}, (b) \{8, 16, 24, \dots\}, (c) \{1, 3, 5, \dots, 99\}, (d) \{\dots, -9, -6, -3\}.
- True or false: (a) 88 \in \{8, 16, 24, \dots, 80\}, (b) 88 \in \{8, 16, 24, \dots\}, (c) 60 \in \{8, 16, 24, \dots\}.
- Write each set as a roster with an ellipsis. (a) The positive multiples of 6. (b) The odd whole numbers from 21 to 59; how many elements does it have? (c) The negative integers.
- Let T = \{10, 100, 1000, \dots\}. Describe T by a rule in words. Then say whether each of 10{,}000{,}000, 500 and 1 is an element of T.
Rules in words
- Describe each set by a rule in words. (a) \{1, 8, 27, 64, 125\}, (b) \{0, -1, -2, -3, \dots\}, (c) \{11, 22, 33, \dots, 99\}.
- A club has six members: Anselma, Barnaby-Jo, Cillian, Desdemona, Eero and Florimel. Write each set as a roster. (a) The members whose names begin with a vowel. (b) The members whose names end in the letter a. (c) The members whose names begin with the letter Z.
The empty set
- Which of these sets are empty? For each one that is not, write it as a roster. (a) The whole numbers between 12 and 13. (b) The solutions of 4x = 0. (c) The solutions of x + 2 = x + 5. (d) The months with exactly 30 days. (e) \{\varnothing\}.
- How many elements does each set have? (a) \{0, 0, 0\}, (b) \{\varnothing, 0\}, (c) \{\{\ \}, \varnothing\}, (d) \{\varnothing, \{0\}\}.
Reasoning
- Someone writes \{3, 5, 7, \dots\}. Give two different rules that both begin 3, 5, 7, with the next entry under each, and name a number whose membership the notation leaves unsettled.
- Explain why “the even prime numbers greater than 2” and “the whole numbers whose square is 3” name the same set.
- For each set, decide whether a complete roster, a roster with an ellipsis, or only a rule in words suits it, and describe it that way. (a) The solutions of x^2 = 100. (b) The numbers greater than 1 and less than 2. (c) The whole numbers less than 500 that are multiples of 9.
- Desdemona says that \{\varnothing\} is empty, “because \varnothing means nothing”. Explain what is wrong with her reasoning.
Answers
- (a) \{32, 34, 36, 38, 40\}. (b) \{-3, -2, -1, 0, 1\}; “strictly” leaves out -4 and 2. (c) \{1, 2, 3, 6, 9, 18\}, from 18 = 1 \cdot 18 = 2 \cdot 9 = 3 \cdot 6. (d) \{\text{January}, \text{June}, \text{July}\}.
- (a) \{-8, 8\}, since 8^2 = (-8)^2 = 64. (b) \{-1\}: the square is 0 only when x + 1 = 0.
- (a), (b) and (d). Order and repeats do not matter, and the multiples of 10 between 5 and 35 are 10, 20 and 30. Set (c) has the extra element 40. Set (e) is \{0, 10, 20, 30\}, since 0 = 10 \cdot 0.
- (a) Finite, 10 elements, from 8 \cdot 1 to 8 \cdot 10. (b) Infinite. (c) Finite, 50 elements: 2k - 1 for k from 1 to 50. (d) Infinite.
- (a) False: 88 is past the last entry, 80. (b) True: 88 = 8 \cdot 11. (c) False: 60 = 8 \cdot 7 + 4 is not a multiple of 8.
- (a) \{6, 12, 18, \dots\}. (b) \{21, 23, 25, \dots, 59\}, with 20 elements, since \displaystyle \frac{59 - 21}{2} + 1 = 20. (c) \{\dots, -3, -2, -1\}.
- T is the set of numbers 10^n with n a positive integer. 10{,}000{,}000 = 10^7 \in T. 500 \notin T, since it is not a power of 10. 1 \notin T: although 1 = 10^0, the list starts at 10.
- (a) The cubes of the whole numbers from 1 to 5. (b) The integers less than or equal to 0. (c) The positive multiples of 11 less than 100, or the two-digit whole numbers whose two digits are equal; both give the same 9 elements.
- (a) \{\text{Anselma}, \text{Eero}\}. (b) \{\text{Anselma}, \text{Desdemona}\}. (c) \varnothing, since no member’s name begins with Z.
- (a) and (c) are empty. (a) 12 and 13 are next to each other. (b) \{0\}, not empty. (c) Subtracting x leaves 2 = 5, false for every x. (d) \{\text{April}, \text{June}, \text{September}, \text{November}\}. (e) Not empty: one element, \varnothing.
- (a) 1, the number 0. (b) 2, a set and a number. (c) 1: \{\ \} and \varnothing are the same set, so this is \{\varnothing\}. (d) 2, the different sets \varnothing and \{0\}.
- “The odd whole numbers from 3 on” gives next entry 9; “the prime numbers from 3 on” gives next entry 11, since 9 = 3 \cdot 3 is not prime. So 9 is in the set under one rule and not the other, and its membership is unsettled until the rule is written in words.
- Both sets are empty. An even number greater than 2 has the factor 2 besides 1 and itself, so it is not prime. And 1^2 = 1 while 2^2 = 4, with larger whole numbers giving larger squares, so no whole number squares to 3. There is only one empty set, so the two descriptions name the same set.
- (a) A complete roster, \{-10, 10\}. (b) Only a rule in words: between any two of its numbers lies another, so there is no next element for an ellipsis to step to. (c) A roster with an ellipsis, \{0, 9, 18, \dots, 495\}, since 9 \cdot 55 = 495 and 9 \cdot 56 = 504; it has 56 elements.
- The set \{\varnothing\} has one element, the empty set itself. The symbol \varnothing names a set with nothing in it, but here that set sits inside braces as an element, like an empty box inside a box. So \varnothing \in \{\varnothing\}, and \{\varnothing\} is not empty.