Properties of Sets

Try this

Tap $18$ , then $12$ : you get $6$ . Now subtract using $6$ and the other elements. Which numbers can you get?
Tap “one round” a few times. What does $S$ end up looking like in $[- 24, 24]$ ? Compare it with the blue ticks.
Is there a way to force $3$ ? Try it, then think about why it fails.
Where does the “Smallest positive element” meter finally stop? How is it related to $12$ and $18$ ?

Given that a set has some property, which elements must it contain, and what other properties must it have? This is the most common type of set problem. The approach is always two steps: first push outward with the rule and see what the set is forced to become; then find a quantity the rule cannot change (here, “divisible by $6$ ”) to show that nothing more can be forced.

Predict first

$S$ is closed under subtraction ( $a, b \in S \Rightarrow a - b \in S$ ), and $21 \in S$ , $35 \in S$ . What is the smallest positive integer that $S$ must contain?

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

Why

A set closed under subtraction that contains 12 and 18

Step 1 / 5

Given
$S$ contains $12$ and $18$ , and is closed under subtraction.
Repeated subtraction
0 and negatives
Every multiple of 6
Nothing more can be forced

Another kind of rule: each element produces the next

A rule can also act on a single element: $a \in S \Rightarrow \frac{1}{1 - a} \in S$ . Start from $2$ and keep applying it. What happens?

Tap a few times. When does the chain come back to where it started?

It goes round in three steps: $a \to \frac{1}{1 - a} \to \frac{a - 1}{a} \to a$ . So as soon as $2 \in S$ , the set $S$ contains at least the three elements $2, - 1, \frac{1}{2}$ . With a different starting point and the rule $a \mapsto \frac{1 + a}{1 - a}$ , how long is the cycle?

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The smallest positive element of $S$ divides every element of $S$

With closure under subtraction, the smallest positive element you end up with, $6$ , divides everything:

Practice

Exercise 1Basic

A set of integers $S$ is closed under subtraction, and $28 \in S$ , $42 \in S$ . What is the smallest positive integer that $S$ must contain?

Exercise 2Basic

A set of real numbers $A$ satisfies $1 \in / A$ and $a \in A \Rightarrow \frac{1}{1 - a} \in A$ . Given that $3 \in A$ , what is the product of the three elements that $A$ must contain?

Exercise 3Advanced

A set of real numbers $B$ satisfies $1 \in / B$ and $a \in B \Rightarrow \frac{1 + a}{1 - a} \in B$ . Given that $2 \in B$ , at least how many elements does $B$ have?

Exercise 4Challenge

A set of real numbers $A$ is non-empty and finite, $1 \in / A$ , and $a \in A \Rightarrow \frac{1}{1 - a} \in A$ . If $∣ A ∣ \leq 10$ , what can $∣ A ∣$ be?

Try this

Why

A set closed under subtraction that contains 12 and 18

Another kind of rule: each element produces the next

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Saying “equals” when you have only shown “must contain”

Thinking S could have just one element