Page 32 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

When a set A ⊆ B but A = B , then A is a proper subset of B , and we can also say “A is properly
contained in B .”

Warning: The Empty Set. There is one special set called the empty set, which we write as ∅. This is the
set which contains no elements. Nothing. Zip. Nada.

∅ = {}.

It is important to remember the empty set, because it can cause sneaky problems in mathematical proofs.
We will use sets to prove theorems, propositions, and lemmas throughout this book, and when we do
this we will often start by proving that a certain set is not empty.

Here are two more things we can do with sets. First, we can combine sets.

Definition 3.1.4 Let A and B be sets. A ∪ B is the union of A and B, which is the set that contains
every element of A as well as every element of B .

We will often write a set based on one or more conditions which must be satisfied by each element of
the set. In general, this way of writing a set looks like

{x such that x satisfies oneormoreconditions}.

The union of sets is similar to adding sets, but, unlike addition, A ∪ A = A, whereas if you add a
number to itself x + x you usually don’t end up with the same number.

Exercise: For which number x is

x + x = x?

Another thing we can do with sets is intersect them.

Definition 3.1.5 Let A and B be sets. A ∩ B is the intersection of A and B , which is the set consisting
of all common elements of A and B ,

A ∩ B = {x such that x ∈ A and x ∈ B}.

For example, {1, 2, 3}∩ {2, 3, 4} = {2, 3} . What is

{1, 2, 3}∩ {4, 5}?

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