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.