Chap 2: Concept of Sets (Part 1)

DEFINITIONS:

①    A set is an unordered collection of objects, known as elements or members of the set.

②    Sets are usually denoted as capital letters; its elements are denoted as lowercase letters.

③    Often, but not always the object in a set have similar properties.

Eg. S = {a, b, c, d, …} (sets of all English alphabets)

       S= {a, 1, ali, law} (random sets)

WAYS OF LISTING ELEMENTS OF SETS:

l  Description

Use a brief sentence to describe the set.

Example: 1. Set S is the set that accepts multiples of 4.

            2. Set D is the set of vowels in English alphabets.

l  Roster Method

Listing the element of a set inside a pair of braces { } that states the members of the set.

Example: 1. The set of odd numbers greater than 15 is written as

                     O= {17, 19, 21, 23, 25,…}

            2. The set V of vowels can be expressed by

               V= {A, E, I, O, U}

l  Set Builder Notation

Used to symbolize a set that characterize all those elements in the set by stating the properties they must have to be its members.

Example: O= {x|x is an odd positive integers less than 10}

SPECIFYING THE PROPERTIES OF SETS:

Finite and Infinite Sets

n  A set is said to be finite if it either contains no elements or the number of the elements in the set is a natural number and presume the existence of the amount.

Eg. 1. A={a, b, c, d}

    2. C={1, 2, 3,…N} N is the amount of element C.

n  A set that is not finite is said to be infinite and counting number is one example of an infinite set.

Eg. 1. B={2, 4, 6,…}

    2. P={a, b, c,…}

Cardinality of Sets

}   It is a measure of the number of elements of a finite set.

}   Cardinal number of set A, symbolized by n(A) or |A| which is the number of elements in set A or the size of set A.

Example: 1. B= {3, 6, 9,} contain 3 elements.

               Therefore, B has a cardinality of 3.

                 

2. G= {ali, abu, ahmad, siti, aminah}

               Set G has a cardinality of 5.

SET MEMBERSHIP:

}  The symbol ∈, read “is an element of” is used to indicate membership in a set.

}  The symbol ∉, read “is not an element of”, is used to indicate that an element does not belong to a set.

}  Example: let A={2, 4, 6,8} and B={1, 3, 5, 7, 9}

                    

                     1. 6 ∈ A

                     2. 5 ∈ B

                     3. 8 ∉ B

                     4. 9 ∉ A

EMPTY SET:

}  Special set that has no element. Also called null set that donated by Ø and { }.

}  Set with one element is called a singleton set, {Ø}.

}  Example: A = {2, 6, 8} and B = {3, 5, 7}
Let C represent the intersection of the sets A and B.
C = {}, a null set, because there is no element common between the two sets.  

    Solved Example on Empty Set

   

Example: Solve C={x| xєR, |6x - 7| + 10 = 0}

Solution:
Step 1: |6x - 7| + 10 = 0
Step 2: |6x - 7| = - 10, which is never true.  [Subtract 10 from each side.]
Step 3: So, the solution set is {} or a null set ø.

p/s: You can get this note at the following link --> Concept Of Sets Part 1