VI. Operation of Set (continuation)

3. Difference of two sets.
If A and B two sets, then we can form a new set of elements consisting of members of set A is not a member of set B. This new set of written A - B, and read "set A less set B"; or if it is written with a set-forming notation : A - B = {x | x ∈ A and x ∉ B}.
Formed a new set of elements may also be a member of set B are not members of set A, and it is written as B - A, read as "less a set A set B"; or if it is written with a set-forming notation: B-A = {x | x ∈ B and x ∉ A}.

For example:
If A = {1, 2, 3, 4, 5, 6, 7} and B = {1, 3, 5}, then A - B = {2, 4, 6, 7}
If A = {a, b, c} and B = {a, b, c, d, e, f}, then B - A = {d, e, f}
If C = {3, 5, 7} and D = {3, 5, 7}, then A-B = {} or A - B the set is empty.

4. The set of the same Set.

Two sets A and B is said to equal if and only if every member of A is a member of B as well, or vice versa, every member of B is a member of A as well.
Two sets A and B are the same written A = B, or if written with a set forming notation set : A = B = {x | x ∈ A and x ∈ B}.
For example:
If A = {2, 3, 1, 5, 4} and B = {1, 2, 3, 4, 5}, then A = B.
If A = {b, c, d} and B = {b, c, d}, then A = B.
Note that the order of the writing member is not a problem on any set (not shown differences in membership of a set).

VII. OPERATION of SET (continuation).

2. Combined sets.

Of the two sets P and Q we can establish a new set of C whose membership consists of the elements of P and Q. The set K is called set combination of P and Q. The set combination is written: K = P ∪ Q, reads: The set of K equal to the combined P Q or set of K equal to the combined P and Q. When written in a way "set shaper": K = {x | x ∈ P or x ∈ Q}.

For example:
a. If A = {p, q, r} and B = {k, l, m, n, o, p, q, r}, then A ∪ B = {p, q, r, k, l, m, n , o}

b. If C = {-1, 0, 1, 2, 3, 4, 6} and D = {0, 2, 4, 6}, then C ∪ D = {-1, 0, 1, 2, 3, 4 , 6}

c. If K = {the boys Mr. Amir} and L = {girls pack Amir
then K ∪ L = {children} pack Amir

d. If P = {2, 3, 5, 7,} and P = {2, 3, 5, 7}, then P ∪ Q = {2, 3, 5, 7,}

The examples above, if drawn with a Venn diagram:
Fig. 5.

VII. OPERATION of SET.

1. Sliced two sets (Intersection)

Of the two sets C and D we can create a new set E whose members include the element C that also includes elements of D. New set of E is called the set of slices C and D.

The set of slices are written C ∩ D, read: the set of slices C and D. So this example can be written E = C ∩ D, reads: The set E is the set of slices C and D.
By writing the shaper set: E = C ∩ D = {x | x ∈ C and x ∈}

With Venn diagram:

                      Fig. 4.     

For example:
If C = {-1, 0, 1, 2, 3, 4, 5} and D = {1, 3, 5, 7, 9}, then E = C ∩ D = {1, 3, 5}
If A = { inhabitant of Jakarta }, and B = { the people who speak English }, then A ∩ B = {the people of Jakarta who speak English }

 

VI. PICTURE OF SET

There is a useful way to express a set with a picture.
Picture or diagram of this kind is called "Venn Diagram". The name is in accordance with the name of the inventor, namely John Venn, a British mathematician who lived during the years 1834 to 1923.
Rules of the image sets with Venn diagrams:

The Universe set is depicted by rectangles. All elements of the universe is drawn as the dots.
Subset of the universe talks depicted by a closed curve that surrounds the dots belonging to its members.
For example:

U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} with a subset K = {3, 7, 5} and L = {2, 4} , his Venn diagram :

                   Fig. 2.

If a member of the set was too much to be drawn as a dot, then the diagram of its members were not described, simply described set form only, or square and a closed curve with the name alone.

For example :

U = {Senior High Schools in Indonesia}, L = { Senior High Schools in Jakarta and }
K = { Senior High School in North Jakarta }, his Venn diagram:

                              Fig. 3.

The use of shading or color in the Venn diagram will generate a clearer diagram.

 

V. SET TYPES (continuation)

4. The Part set (or subset), namely the set whose members are part of another set member.
For example:
D = {0, 1, 2, 3, 4, 5, 6, 7} and
E = {3, 2, 6, 7}, it appears here that every element E is also a member of the set D.

It says H is "subset" of C and E ⊂ K be written.
"Subset" is denoted by ⊂.
Or say also set D "contains the set of" E and is written D ⊃ E. "Contains a set" is denoted by ⊃.
The set E is a subset of D if each member / element E is also a member of D. And because each member of E is contained in D, then E is said also contained in D.

How many subsets of a set?

From the set K = {2, 3, 5} we can form subsets: {}, {2}, {3}, {5}, {2, 3}, {2, 5}, {3, 5} and {2, 3, 5}.
Subsets obtained by 8 = 2³ (2 to the 3rd), 3 is the number of elements of K.

Thus the number of subsets of the set which has n element is 2ⁿ (two raised to the n th power).

Note: The empty set (null set) is a subset of all kinds of sets.
5. The Same set, namely two sets of all its members together.
For example:
If K = {a, b, c, d} and L = {b, d, a}, then is said to set K = set of L and is written K = L.
Note: The order of the members in each set not to be the same.

K and L is said to equal if every element of K is an element of L and vice versa

every element of L is an element of K as well.

6. The Universe set or the Universal set or Universum, namely a set that contains all objects that are being discussed.
The objects that being discussed had established a set of the Universe.
The Universe set is represented by "U" (from Universum / Universal). On the set of Venn diagrams, the Universe set depicted with rectangular.
Example:
             If A = {0, 2, 4, 6}, then set A is probably derived from
U = {integers}
U = {whole numbers}
U = {even numbers less than 8} or the set of another universe

containing 0, 2, 4, and 6.
Obviously when we talk about a set, of course this set is a subset of a specific universe set.

V. SET TYPES

1. The Finite sets, which sets has the number of members is limited (finite).
          For example:
                    A = {1, 3, 5, 7}, set A is finite set because n (A) = 5.

2. The Infinite sets, namely that the number of members is infinity (infinite).
           For example:
             A = {1, 2, 3, 4,. . . }, Set A is infinite set because n (A) = infinity.

3. The Empty set (empty set or void or NULL set), is set that does not contain anything that element.
For example:
         B = {four-legged chickens}, and
         C = {x | x < 7, divisible by 9, x natural numbers}.
It is clear that there is no chicken-legged 4 and obviously also no natural numbers which satisfy the required conditions on the set C. Therefore, set B and C above do not contain elements and said something that set B and C is the set of "empty".

Empty set is written with the symbol ∅ or { }.
So B = ∅ or B = { } and C = ∅ or C = { }

III. HOW TO REPRESENT AN ELEMENT / MEMBER OF SET

How do I write or declare an object or something as a member / element of a specific set?
To say "member" of a set of used notation "∈". To say "not a member" used the notation "∉".
For example:

If A = {2, 3, 5, 7, 11}, then:
2 contained in A, means that 2 members of A and written 2 ∈ A
7 contained in A, means the 7 members of A and written 7 ∈ A
1 is not contained in A, means 1 is not a member of A and written 1 ∉ A
6 is not contained in A, means that 6 is not a member of A and written 6 ∉ A
etc.

IV. ELEMENTS OF A SET OF NUMBER PASSED
To state the amount or number of members of the set used the notation "n (name set)"
For example:
A = {chicken, ducks, goats, cows}, the number of members of A = 4, and write n (A) = 4.
B = {2, 4, 6, 8, 10}, the number of members of B = 5, and is written n (B) = 5.

II. HOW TO REPRESENT A SET OF

As a symbol of a set of use curly braces {...}. Statements about members of the set were written between curly braces open and close it.

To distinguish a set with another set, a set is usually expressed (given name) with an uppercase (capital), for example: A, B, C, D,. . . , Or Z.

How do I declare or write a Set ?

There are 4 (four) way of writing to express a set:

1. By way of registering or tabulation, in particular by mentioning / write members one by one. Writing members to each other separated by a comma.

1.a. For a set with members of the limited and small. Members of the set is written all one by one.
For example:
A = {-1, -2, -3, -4, -5}
B = {January, February, March, April}

1.b. For the set with a limited member but many. Members may not be listed all. Its members are listed some of course, continue with 3 points. . . , (Mean / read and so on) and then ends with the last member.
For example:
C = {1, 2, 3, 4, 5,. . . , 69}
D = {a, b, c, d, e,. . . , z}

1.c. For sets with many members and is not limited. Members of the set is represented by some only, at least 4, and for other members who are not mentioned enough to be represented by 3 points. . . , Which means and so on.
For example:
E = {1, 2, 3, 4,. . . }
F = {Bandung, Jakarta, Semarang, Bogor. . . . }

2. Description formula, namely with mention membership requirements.
For example:
Example in 1.a above, if written in a description formula:
A = { negative interger between 0 and -6}
B = {name of first 4 months of the year AD}

3. By writing the member notation of the set.

3.a. Every object or object member is represented by a set of variables.
For example:
The set B above when written this way to be B = {a, b, c, d}. So, a representative of January, b represents February, March and d c represents a representing March.

3.b. All members of the set denoted by a notation and write "membership requirements" behind the sign "|".
For example:
If the sample 1.a. above was written this way to be:
A = {x | 0> x> - 6, x integer}
B = {x | x = name of the first 4 months of the year AD}

4. With Picture of sets. Venn Diagran
For example: Venn diagram of A = {1, 2, 3, 4, 5, 6}
Fig. 1.

A. SET CONCEPT

1. UNDERSTANDING
An understanding of the set is one of the important and fundamental concepts in modern mathematics. The concept of the set was first introduced officially by a German mathematician Georg Cantor in the 19th century.

The concept of the set is found, explicitly or implicitly, in every field of pure and applied mathematics. Explicitly, the principles and terminology set used to make the statement more clear and precise mathematics and to explain concepts such as limited and unlimited.

What is the set?

SET is a collection of objects or anything else that is clearly defined so that we can easily find out whether an object or something else was included in a set or not.

The objects or other things that are included in a set called members or elements of the set or collection of these.
If someone says: "Above the dining table is a stack of notebooks." So we can imagine that on the dining table was a stack of notebooks, in the pile notebook only, not other objects, papers or other.

In a sentence on a stack is a set and a member or set of elements that are writing books, writing books and not just reading books or other objects.
In the set of the above a member conditions are notebooks are piled on the dining table, and from the membership requirement is to clearly know where the objects belonging to members of the set which do not include members of the set.

So it is clear to create a Set, the membership requirements of the members or elements of a set that will be formed must be presented clearly so that for any set anyone can tell whether an object or certain other things that are members of the set or not.

Terms or the nature of the set of member characteristics on a set. Characteristic is what distinguishes a set with another set. These characteristics are also useful for giving a name to a set.

Another example:
The set of integers. Here the member characteristics / requirements of a set are integers.
Association of European countries. Here the member requirements / characteristics are the countries in continental Europe
Etc..