Talking about sets, then we do not need to go far to imagine it, because in life and in our environment there are many things related to data (objects) that are collected based on certain criteria. This collection of data will later be defined as a set. In addition, sets also have several types.
Weldebis Association
A set is a collection of distinct objects that can be clearly defined. Objects in a set are called elements or members of the set. Membership of a set is indicated by the notation '∈'.
Example 1:
- A = {x, y, z}
- x ∈ A : x is a member of the set A.
- w ∉ A : w is not a member of set A.
There are several ways to express a set, namely:
a. Enumerating its members (enumeration)
In this way, the set is stated by mentioning all the members of the set within a curly bracket.
Example 2:
- The set of the first four odd numbers: A = {1, 3, 5, 7}.
- The set of the first five prime numbers: B = {2, 3, 5, 7, 11}.
- The set of natural numbers less than 50 : C = {1, 2, ..., 50}
- The set of integers is written as {..., -2, -1, 0, 1, 2, ...}.
b. Using standard symbols
A set can be expressed in a standard symbol that is generally known by the (scientific) community.
Example 3:
- N = set of natural numbers = { 1, 2, ... }
- Z = set of integers = { ..., -2, -1, 0, 1, 2, ... }
- Q = set of rational numbers
- R = set of real numbers
- C = set of complex numbers
The universal set (universe of discourse) is denoted by U.
Example 4:
Suppose U = {1, 2, 3, 4, 5} and A = {1, 3, 5} are subsets of U.
c. Writing the criteria (requirements) for membership of the set
A set can be expressed by writing the criteria (requirements) for membership of the set. This set is denoted as follows:
{ x | conditions that must be met by x }
Example 5:
(i) A is the set of natural numbers less than 10, A = { x | x ≤ 10 and x ∈ N } or A = { x ∈ N | x ≤ 10 } which is equivalent to A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(ii) M = { x | x is a student taking a discrete mathematics course}, or M = { x is a student | he is taking a discrete mathematics course}
d. Using Venn Diagram
A set can be expressed by writing its members in a picture (diagram) called a Venn diagram.
Example 6:
Let U = {1, 2, ..., 7, 8}, A = {1, 2, 3, 5} and B = {2, 5, 6, 8}.
Venn Diagram:
Venn Diagram
Regarding the issue of membership, a set can be expressed as a member of another set.
Example 7:
a. For example, M = { STMIK EL RAHMA students }
- M1 = {student members of himael}
- M2 = {student members of HMTI}
- M3 = {student members of HMSI}
Thus, M = { M1, M2, M3 }
b. If P1 = {x, y}, P2 = { {x, y} } or P2={P1},
- Meanwhile, P3 = {{{x, y}}}, then x ∈ P1 and y ∉ P2,f
- so that P1 ∈ P2 , while P1 ∉ P3, but P2 ∈ P3
The number of elements in a set is called the cardinality of the set. For example, to express the cardinality of set A, it is written with the notation:
n(A) or |A|
Example 8:
- B = {x | x is a prime number smaller than 10},
- or B = {2, 3, 5, 7} then |B| = 4
- A = {a, {a}, {{a}} }, then |A| = 3
If a set has no members, in other words the cardinality of the set is equal to zero, then the set is called an empty set (null set). The notation for an empty set is: ∅ or {}.
Example 9:
- P = {Industrial Engineering students of STT Telkom who have been to Mars}, then n(P)= 0, So P= ∅
- A = {x | roots of the quadratic equation x2 + 1 = 0 and x ∈ R}, then n(A) = 0, So A = {}
- B = {{ }} can also be written as B = {∅}, so B is not an empty set because it contains one element, namely the empty set.
A set A is said to be a subset of a set B if and only if every element of A is an element of B. In this case, B is said to be a superset of A. Subset notation: A ⊆ B or A ⊂ B
If depicted in the form of a Venn diagram, the subset becomes:
Venn diagram of subsets
Example 10:
- (i) N ⊆ Z ⊆ R ⊆ C
- (ii) {2, 3, 5} ⊆ {2, 3, 5}
For each set A the following applies:
- (a) A is a subset of A itself (i.e., A ⊆ A).
- (b) The empty set is a subset of A ( ∅ ⊆ A).
- (c) If A ⊆ B and B ⊆ C, then A ⊆ C
∅ ⊆ A and A ⊆ A, then ∅ and A are called improper subsets of set A. The statement A ⊆ B is different from A ⊂ B : A ⊂ B : A is a subset of B but A ≠ B. Thus, A is a proper subset of B.
Example 11:
Let A = {1, 2, 3}.
{1} and {2, 3} are proper subsets of A.
The power set of a set A is a set whose elements are all subsets of A, including the empty set and the set A itself. The power set is denoted by P(A). The number of elements (cardinals) of a power set depends on the cardinals of the original set. For example, the cardinality of set A is m, then |P(A)| = 2m.
Example 12:
If A = { x, y }, then P(A) = { ∅, { x }, { y }, { x, y }}
Example 13:
The power set of the empty set is P(∅) = {∅}, while the power set of the {∅} set is P({∅}) = {∅, {∅}}.
The statement A ⊆ B is used to state that A is a subset of B for which A = B is possible.
Two sets are said to be equal if they satisfy the following conditions:
- A = B if and only if every element of A is an element of B and vice versa every element of B is an element of A.
- To state A = B, what needs to be proven is that A is a subset of B and B is a subset of A. If this is not the case, then A ≠ B. or A = B <=> A ⊆ B and B ⊆ A.
Example 14:
- If A = { 0, 1 } and B = { x | x (x -- 1) = 0 }, then A = B
- If A = { 3, 5, 8, 5 } and B = {5, 3, 8 }, then A = B
- If A = { 3, 5, 8, 5 } and B = {3, 8}, then A ≠ B
For three sets, A, B, and C, the following axiom applies:
- (a) A = A, B = B, and C = C
- (b) If A = B, then B = A
- (c) If A = B and B = C, then A = C
Two sets are said to be equivalent if each has the same cardinality. For example, set A is equivalent to set B, meaning the cardinals of set A and set B are the same, the notation used is: A ~ B
Example 15:
Suppose A = { 2, 3, 5, 7 } and B = { a, b, c, d }, then A ~ B because |A| = |B| = 4
Two sets A and B are said to be disjoint if they do not have any elements in common. The notation used is A // B . If expressed in the form of a Venn diagram it is as follows:
Disjoint Set
Example 16:
If A = { x | x ∈ N, x < 10 } and B = { 11, 12, 13, 14, 15 }, then A // B.