1. 1.1 Sets and Subsets

2. A set is a well defined collection of objects
A collection of beanie babies
A collection of hats
An Element (∈) is one of the objects in a set
A = {1, 2, 3}
1 ∈ A
2 ∈ A
3 ∈ A
4 ∉ A

3. We specify a property when it is difficult to list all elements.
We write this in the form {x|P(x)}
(x such that P of x)
A = {1,2,3} could be written as:
{x|x is a positive integer less than 4}

4. Order in a set does not matter
{1,2,3}
{3,2,1}
Special sets:
ℤ(zeta) integers {..., −3, −2, −1, 0, 1, 2, 3, ...}
ℕ(nu) natural numbers { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}.
ℚ Rational numbers integers, fractions and decimals
ℝ Real numbers Rational and Irrational numbers (square root )
{}Empty set

5. Equality
Two sets A and B are equal if they have the same elements:
If A= {Java, Pascal, c++}
and
B = {Pascal, c++, Java}
then A=B

6. Subsets ⊆ is contained in
A ⊆ B
A={1,2,3,4}
B = {1,2,5,6}
A is not contained in B

7. Venn Diagrams
The Venn diagram shows relationships between sets.
of what we are talking about
A = a set of beanie babies
B = a set of hats
The intersection ∩ of A and B,
written A ∩B are beanie babies with hats.

8. The Power Set of A , written P(A), is the set of all possible subsets of A.
Note there are 3 elements
23 = 8
There are 8 possible subsets
P(A) = {}, {1}, {2}, {3}, {1,2},{1,3},{2,3},{1,2,3}