A mason announces: “I build a house only for people who do not build their own.” Does he build his own? Whichever way you answer, the rule breaks. That puzzle hides a real question: when is a collection a set? The answer waits at the end of the note. First, the tools for writing sets.
1. A set and its elements
Any well-defined collection of objects of the real world or of the conceptual realm is called a set. Well defined means that for any object we can say for sure whether it is in the collection or not.
Each of the objects forming a set is an element of that set. If x is an element of A, we write x ∈ A. If it is not, we write x ∉ A.
2. Two ways of writing a set
In the tabular method the elements are listed inside { }. The set of squares of the natural numbers not greater than 10 is S = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}.
In the set builder method the set is described by the common property of its elements. The same S is S = {x : x is a square number not greater than 100}.
So there are two methods of writing a set: the tabular method and the set builder method. {1, 2, 3, 4}, for example, is written in the tabular method. Sort six written sets below.
which method?chip, then bin
Tabular or set builder
Pick up a chip, then drop it in a bin. Tabular means the elements are written out.
Right. Next one.
Not yet. Inside { }, are the elements written out, or is there an x and a condition?
All in place. Writing the elements out is the tabular method; describing them by a common property is the set builder method.
3. From a condition to a list
To go from set builder to a list, test the condition against every element. With U = {1, 2, ···, 10}: {x ∈ U : 5x > 37} = {8, 9, 10}, {x ∈ U : x + 5 < 12} = {1, 2, 3, 4, 5, 6}, {x ∈ U : 6 < 2x < 17} = {4, 5, 6, 7, 8}, and {x ∈ U : x² < 37} = {1, 2, 3, 4, 5, 6}. Pick a condition yourself and see which tiles light up.
pick a conditionchange the condition
Condition to list
The tiles are drawn by hand, not to scale. A lit tile is an element.
In Exercise 1.1 No.5, U = {x : 1 ≤ x ≤ 20, x ∈ Z}. The odd numbers in it form A = {1, 3, 5, ···, 19}, and the primes form B = {2, 3, 5, 7, 11, 13, 17, 19}.
4. The universal set
In a discussion about sets, if the elements of all the sets under discussion belong to one particular set, that set is called the universal set. It is written U.
For instance S = {x : x is a positive integer and 5x ≤ 16} = {1, 2, 3}, T = {x : x is a positive integer and x² < 20} = {1, 2, 3, 4}, and P = {x : x is a positive integer and √x ≤ 2} = {1, 2, 3, 4}. A universal set for all three is U = {x : x is a positive integer}.
5. The number sets: N, Z, Q, R
N = {1, 2, 3, ···} is the set of all natural numbers, or positive integers. Z = {···, −2, −1, 0, 1, 2, 3, ···} is the set of all integers.
Q = {x : x = p/q, where p is any integer and q is any positive integer} is the set of all rational numbers. Since q is positive it can never be 0, which is why writing Q = {p/q : p, q ∈ Z} is wrong. Adding the condition, Q = {p/q : p, q ∈ Z and q ≠ 0} is correct. The set of positive real numbers is written R₊ (or ℝ⁺).
The four sets sit inside one another like boxes. Drop a number in and it falls through every box it belongs to.
drop a numbera number, then a guess
Which box does it stop in?
The boxes are drawn by hand, not to scale. Box size does not mean set size: all four are infinite.
R = {x : x is a real number} is the set of all real numbers. Every natural number is an integer, every integer is rational and every rational is real. So N ⊂ Z ⊂ Q ⊂ R. Put them in order, smallest to largest.
smallest to largesttap in order
Chain of number sets
Start with the smallest set.
Yes. Which comes next?
Not yet. −1 is an integer but not natural, so Z is bigger. 1/2 is rational but not an integer, so Q is bigger still. √2 is real but not rational, so R is the largest.
N ⊂ Z ⊂ Q ⊂ R. Every natural number is an integer, every integer is rational, every rational is real.
6. When a collection is not a set
A collection is a set only if it is well defined. Back to the mason. If he builds his own house he breaks his rule, and if he does not, the rule says he must build it. In the same way, for A = {x : x ∉ A}, assuming A ∈ A gives A ∉ A, and assuming A ∉ A gives A ∈ A. Such a description gives no well-defined collection, so it is not a set. An empty collection, though, is well defined, so it is a set. Tap the seven descriptions and see.
spot the settap a chip
Is the collection well defined?
A set. We can say for sure whether any number belongs.
A set. Its elements are 1, 2, 3.
A set. It is the empty set, still well defined.
A set. The empty set, as of 2014 in the book.
Not a set. Both answers contradict the rule.
Not a set. A ∈ A forces A ∉ A, and the reverse.
A set. That is Q, well defined.
All seven seen. Well defined means a set, even when empty. A description that contradicts itself is not a set.
7. Check yourself
The first question is right from the basics.
check yourselfpick one answer
Question 1
Which relation among the sets of natural numbers (N), integers (Z), rational numbers (Q) and real numbers (R) is correct?
N is the smallest set and must come first; Q ⊂ N fails because 1/2 is not natural. The chain is N ⊂ Z ⊂ Q ⊂ R.
R is the largest set and must come last; R ⊂ Q fails because √2 is not rational.
Right. Every natural number is an integer, every integer is rational and every rational number is real.
Q ⊂ N is false: 1/2 is rational but not a natural number. The chain is N ⊂ Z ⊂ Q ⊂ R.
In the second, watch the symbol.
check yourselfpick one answer
Question 2
Which set is S = {x : x ∈ R and x(x − 2) = x² − 2x} equal to?
x(x − 2) always expands to x² − 2x, so no x fails the condition; the set is R, not empty.
The condition holds for every real number, not only the natural ones, and x ∈ R is given, so S = R.
Non-integers such as −1/2 also satisfy the condition, so S is the whole of R.
Right. x(x − 2) = x² − 2x is an identity, true for every real x, so S = R.
The third is about testing a condition.
check yourselfpick one answer
Question 3
Which of the following statements are correct?
i. If a set has 2n elements, it has 4ⁿ subsets ii. Q = {p/q : p, q ∈ Z and q ≠ 0} is the set of all rational numbers iii. For a, b ∈ R, (a, b) = {x ∈ R : a < x < b}
iii is also true: (a, b) is the open interval {x ∈ R : a < x < b}, so all three statements hold.
ii is true too: with q ≠ 0, p/q gives every rational number, so all three statements hold.
i is true too: 2n elements give 2²ⁿ = 4ⁿ subsets, so all three statements hold.
Right. i: 2²ⁿ = 4ⁿ; ii: with q ≠ 0 this is the set of rational numbers; iii: the definition of an open interval.
The whole note on six cards.
cut from the notetap to turn over
Six cards
QuestionHow does the book define a set?
AnswerAny well-defined collection of objects of the real world or of the conceptual realm.
QuestionWhat is the method called when a set is written by listing its elements, like {1, 4, 9, 16, 25}?
AnswerTabular method
QuestionWhat is the method called when a set is written as {x : x is a square number not greater than 100}?
AnswerSet builder method
QuestionWhat is the particular set called that contains the elements of all the sets under discussion?
AnswerThe universal set (U)
QuestionThe chain of number sets: N ⊂ … ⊂ Q ⊂ R
AnswerZ
QuestionWhy can A = {x : x ∉ A} not be called a set?
AnswerAssuming A ∈ A gives A ∉ A and the reverse; the collection is not well defined.
Source: Higher Mathematics, Classes 9-10 (NCTB), Chapter 1 “Set and Function”, the “Sets and Ways of Writing Them” part.