Elements and Sets

Try this

Without revealing the answers, decide about each of $1 \sim 12$ . Which ones can you write as $a^{2} - b^{2}$ at a glance?
Tap “Reveal”, then “Rows by residue mod $4$ ”. One whole row contains no element of $S$ at all. Which row?
Tap a number that is not in $S$ and read why it is not.
Now tap a number that is in $S$ . Is the machine's way of writing it the only one? For example, as which other difference of two squares can you write $15$ ?

The set $S = {x ∣ P (x)}$ means: the elements of $S$ are exactly the objects with property $P$ (the comprehension principle). Deciding whether $x \in S$ means deciding whether $x$ has property $P$ . If the property is stated clearly, the set is well defined. But “stated clearly” is not the same as “easy to decide”: that empty row above has to be proved, not just seen.

Predict first

$S = {x ∣ x = a^{2} - b^{2}, a, b \in Z}$ . Does $2026$ belong to $S$ ?

YesNoToo big to tell

Choose an answer first; then you can see the result.

Why

What we have to prove is that two sets are equal: $S = {x \in Z ∣ x does not leave remainder 2 on division by 4}$ . The description on the left is hard to check; the one on the right can be checked at a glance.

Which numbers are a difference of two squares?

Step 1 / 5

Every odd number is in
An odd number $2 k + 1$ is the difference of two consecutive squares. The rows for remainder $1$ and remainder $3$ lie entirely in $S$ .

$2 k + 1 = (k + 1)^{2} - k^{2}$
Every multiple of 4 is in
Squares mod 4
A difference never leaves 2
Two descriptions, one set

Change the property: sums of squares

Swap “difference” for “sum”: $T = {x ∣ x = a^{2} + b^{2}, a, b \in Z}$ . Sort the numbers by their remainder mod $4$ again.

Which row is empty? Are all the other rows full?

The row for remainder $3$ is empty, for the same reason as before: the sum of two squares can only be $0, 1, 2$ mod $4$ . But this time the converse is false: $21$ and $33$ leave remainder $1$ , yet neither is a sum of two squares. “Not leaving remainder $3$ mod $4$ ” is only a necessary condition for belonging to $T$ ; it cannot serve as another description of $T$ .

Three properties of elements

The elements of a set have three properties, and each one can bite you when solving problems:

Definiteness: for any given object, it is either in the set or not, one or the other. “Very large numbers” do not form a set.
Distinctness: each element is counted only once. ${1, 2, 2}$ is just ${1, 2}$ , a two-element set.
Unorderedness: ${1, 2, 3} = {3, 1, 2}$ .

The number of elements of $A$ is written $∣ A ∣$ . Because of distinctness, a set like ${x + y ∣ x, y \in A}$ often has far fewer elements than there are “combinations”. The exercises below are about counting how many are left once repeats are removed.

Practice

Exercise 1Basic

How many of $1, 2, \dots, 100$ can be written as the difference of two integer squares?

Exercise 2Basic

Let $A = {1, 3, 4, 8}$ and $B = {x - y ∣ x \in A, y \in A}$ . Find $∣ B ∣$ .

Exercise 3Advanced

Given that $1 \in {a, a^{2}, 3 a - 2}$ and that this set has three elements, find $a$ .

Exercise 4Challenge

How many of $1, 2, \dots, 100$ can be written as $a^{2} - b^{2}$ with $a, b$ both positive integers?

Try this

Why

Which numbers are a difference of two squares?

Change the property: sums of squares

Three properties of elements

Practice

Mistake Museum · Exhibits from this stop

Solving for the parameter and forgetting to check distinctness

Treating ∅ and {∅} as the same thing