A. Important Sets in Mathematics
1. N = {x: x is a natural number} = {1, 2, 3, …}
2. R = {x: x is a real number }
3. Z = {x: x is an integer} = {… -3, -2, -1, 0, 1, 2, 3, …}
4. Q = {x: x is a rational number} = {terminating decimal or repeating decimal written as .} examples:
5. W = {x: x is an integer greater than 0} = {0, 1, 2, 3, …}
6. I = {x: x is an irrational number} = {non-repeating decimal, non-ending decimal} examples:
7. Imaginary = {}
B. Set Notation:
{ } is the empty set also known as the null set
Union Set: the set that contains ALL of the elements in set A and Set B.
written
Intersection Set: the set that contains only the element in both set A and Set B.
written
Complementary Set: all the elements that are not in the original set:
written example:
what is the complement of the even numbers? The odd numbers
What is the complement of negative integers? Whole numbers
A – B = whatever is in set A minus any of the elements in Set B
Example: Set A = {3, 5, 7} and Set B = {3, 4, 6} so A – B = {5, 7}
Another Example:
Let A = even #’s and B = multiples of 3 both set A and B are from 1 to 15
A = {2, 4, 6, 8, 10, 12, 14}
B = {3, 6, 9, 12, 15}
Recall that (x, y) is an ordered pair.
{x, y} is a set that contains the elements of x and y.
We will be using Venn Diagrams to show sets.
This is written as {x, y} Set A
Notation means contained in.
This is written as {1, 3, 5} Set B
C. Set Identities:
1. Commutative Law:
Example: Set A = {a, b} and Set B = {1, 2, 3}
and so they are same.
2. Distributive Law:
3. De Morgan’s Law:
4. Complement of Set A
Example:
Let A = {a, b, c} and B = {d, e, f}
Homework: using the following sets, A = {1, 2, 3}, B= {2, 3, 4} and C = {1, 3, 5}
Show: the commutative property using sets A and B
Show: The distributive property using all three
Find the following: