Mathematics for computing (Introduction to sets)

 Introduction to sets

             sourse:https://www.computersciencedegreehub.com/wp-content/uploads/2020/05/Does-a-Computer-Science-Degree-Require-A-Lot-of-Math-Courses-768x432.jpg

We will discuss the below topics;

• Definition of
  set
• Properties of sets
• Elements
• Specifying a set
• Some common sets 
• Equality of two sets
• Cardinality of sets
• Finite sets
• Infinite sets
• Complement of a sets

Definition of a set

    A set is an unordered collection of zero or more distinct well-defined objects.

            ex: A={1, 2, 3}

                  B={numbers less than 10}

                  C={all positive numbers}

Properties of a set

    Sets are inherently unordered.

    The order in which the elements are presented in a set is not important.

                    A={a, e, i, o, u}

                    B={i, o, u, e, a} both define the same set

    All elements are distinct(unequal).

    One member does not appear more than one time.

            F={a, i, o, u, e, a} is not a set since the element 'a' repeats.

Elements

    The item contained in a set are called elements or members of the set .

    Notation

        - ∈ means " is an element of"

        - ∉ means " not an element of"

    S={x, y, z}

        x ∈ S

        p ∉ S

Specifying a set

    There are three main ways to specify a set.

                    

       # Roster form (listing notation):

                ex: set of even numbers less than 9 = {2, 4, 6, 8}

       # Statement form(descriptive form):

               ex: A={set of even numbers less than 9}

        # Set builder form:

                 ex: A={x: x=2n, n ∈ N and 0<n<5}

Some common sets

Z

    Z is the set of all integers 

    Z={...,-2, -1, 0, 1, 2,...}

N

    Set of the non-negative integers

    N={0, 1, 2, 3,...}

P

    Set of positive integers 

    P={1, 2, 3,...}

Q

    Set of rational numbers 

    Q={a/b: a,b integers, b≠0}

R

    Set of real numbers consisting of integers, rational numbers like -3/4, 22/7, and irrational            numbers like √2, π ...

C

    Set of complex numbers 

Equality of two sets

If  set "A" is equal to set "B"(A=B);

    Both sets have the same elements.

        ex: A={1, 2, 3, 4, 5}

              B=(x: x<6, x ∈ P)

              C={1, 2, 3, 4}

              A=B and A≠C

Cardinality of sets

    Cardinality refers to number of elements in a set. Let "A" be any set then the cardinality of "A"     is denoted by |A|

                ex: A={1, 2, 3, 4, 5}

                      then |A|=5

Finite set

    A finite set is a set that has a finite number of elements. It is countable.

                 ex: A={1, 2, 3, 4, 5}

Infinite set

    An infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.

                ex: A={all the negative numbers}

                      B={all real numbers}

Complement of a set 

The complement of a set is the set that includes all the elements of the universal set that are not present in the given set.

    This is denoted by À / Ā ...

                 ex: y={x: x≥0}

                       ý ={x: x<0}