ITM 107 Lecture Notes - Lecture 1: Natural Number, Empty Set, Null Set

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KATIE CHOI
9/10/18
LECTURE 1 ALGEBRAIC CONCEPTS
Sets
A set is a well-defined collection of objects
There are two ways to tell what a given set contains:
1. Listing the elements (or members) of the set (usually between braces)
o We may say that a set A contains 1, 2, 3, and 4 by writing A = {1,2,3,4}
o To say that 4 is a member of set A, we write 4 A.
o Similarly, we write 5 A to denote that 5 is not a member of set A.
o If all the members of the set can be listed, the set is said to be a finite set. A = {1,
2, 3, 4} and B = {x, y, z} are examples of finite sets.
o For an infinite set, we cannot list all the elements, so we use the three dots. For
example, the set of all whole numbers N = {1, 2, 3, 4, . . .} is an infinite set. This
set N is called the set of natural numbers
2. Another way to specify the elements of a given set is by description.
o For Example
F = {y: y is an odd natural number}
is read “F is the set of all y such that y is an odd natural number.”
Example 1 Describing Sets
o Write the following sets in two ways.
a. The set A of natural numbers less than 6
b. The set B of natural numbers greater than 10
c. The set C containing only 3
o Solution:
a. A = {1, 2, 3, 4, 5} or A = {x: x is a natural number less than 6}
b. B = {11, 12, 13, 14, . . .} or B = {x: x is a natural number greater than 10}
c. C = {3} or C = {x: x = 3}
It is possible for a set to contain no members. Such a set is called the empty set or the
null set, and it is denoted by Ø or by { }.
Special relations that may exist between two sets are defined as follows.
Relations between Sets
o Definition
1. Sets X and Y are equal if they contain the same elements
2. A is called a subset of B, which is written 𝐴 ⊆ 𝐵 if every element of A is
an element of B. The empty set is a subset of every set. Each set A is a
subset of itself.
3. If C and D have no elements in common, they are called disjoint.
o Example
1. If X = {1, 2, 3, 4} an Y = {4, 3, 2, 1}, then X = Y
2. If A = {1, 2, c, f} and B = {1, 2, 3, a, b, c, f}, then A B. Also, A, B, A
A, and B B.
3. If C = {1, 2, a, b} and D = {3, e, 5, c}, then C and D are disjoint
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