Page 30 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Definition 3.1.2. A set S is a subset of a set B if every element of S is also an element of B. We use
the notation

S ⊆ B,

and we say “S is a subset of B.” We may also write

B ⊇ S,

which means “the set B contains the set S as a subset.”

Note that S ⊆ B is equivalent to B ⊇ S .

Exercise: Practice writing S ⊆ B in different places and directions but always with the same meaning.
Does this remind you of an exercise in the previous chapter?

An example is:

{1, 2}⊆ {2, 1, 3, 4},

and

{3, 1, 2, 4}⊇ {2, 1}.

Another example is:

{1, 2}⊆ {1, 2}.

Yes, that’s right, a set is a subset of itself. So, when a set is a subset of another set, the subset is smaller
but it could also be the same. If we think about comparing numbers, the symbol ≤ that we use to
compare numbers is similar to the symbol ⊆ that we use to compare sets. If we have two numbers x
and y , and we can show that x ≤ y and y ≤ x , then, what do we know about the numbers x and
y ? That’s right, they are the same number: x = y . So, if we started with two numbers, and we actually
wanted to show they are the same number, one way to do this would be to show

x ≤ y,

and then to show

y ≤ x.

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