Page 33 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

When sets have no elements in common, then their intersection is the empty set. So,

{1, 2, 3}∩ {4, 5} = {} = ∅.

Intersection and union satisfy the following properties.

1. Commutative: A ∪ B = B ∪ A , and A ∩ B = B ∩ A.
2. Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C), and (A ∩ B) ∩ C = A ∩ (B ∩ C).
3. Distributive: A ∩ (B ∪ C)= (A ∩ B) ∪ (A ∩ C), and
A ∪ (B ∩ C)= (A ∪ B) ∩ (A ∪ C).

Although a set does not have any order, it can be useful for us to assign a number to each element of a
set; this is called indexing the set. For a set S which contains five elements, we pick some element and
. Then we pick a different element, which we call a 2 , and we continue until each element has
call it a 1
been assigned an index. Then we can write

S = {a 1 ,a 2 ,a 3 ,a 4 ,a 5 }.

Definition 3.1.6. Let S be a set which contains k elements, where k is a positive whole number. Then, we
can index the set S by assigning a whole number between 1 and k to each element of the set, and we write

S = {a 1 ,a 2 ,...,a k }.

The notation

k
{a i } i=1

means

{a 1 ,a 2 ,...,a k }.

The notation a i means the i element of the set S, and i is called the index. The notation
th

# S

means the number of elements in the set S .

Sometimes we might want to remove elements from a set. For example, if

S = {a 1 ,a 2 ,a 3 ,a 4 ,a 5 },

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