There are several set operations that need to be known, namely: intersection, union, complement, difference and common difference.
Getting to Know Set Operations
1. Intersection
The intersection of two sets is denoted by the symbol '∩'. Suppose A and B are disjoint sets, then A ∩ B = { x | x ∈ A and x ∈ B }
If expressed in Venn diagram form it is:
intersection
Example of a slice:
- Suppose A = {2, 3, 5, 7, 11} and B = {3, 6, 9, 12}, then A ∩ B = {3}
- Suppose A is a set of IT female students at STT Telkom and B is a set of elderly women (50 years and above), then A ∩ B = ∅.
This means that A and B are mutually exclusive or A // B.
2. Union
The union of two sets is denoted by the symbol '∪'. Suppose A and B are sets, then A ∪ B = { x | x ∈ A or x ∈ B }
Union
If expressed in Venn diagram form it is:
Union example:
- If A = { 2, 3, 5, 7} and B = { 1, 2, 3, 4, 5 }, then A ∪ B = { 1, 2, 3, 4, 5, 7}
- A ∪ ∅ = A
3. Complement
The complement of a set is the elements that are in the universal set (universe of discourse) except the members of the set. Suppose A is a set that is in the universe of discourse U, then the complement of the set A is denoted by:
A = { x | x ∈ U and x ∉ A }
If expressed in Venn diagram form it is:
Complement
Examples of complements:
- Let U = { 1, 2, 3, ..., 9 },
- if A = {1, 3, 7, 9}, then Ā = {2, 4, 5, 6, 8}
- if A = { x ∈ U | x is divisible by two }, then A= { 1, 3, 5, 7, 9 }
Examples of complements:
- A = STT Telkom student association
- B = group of students living in the dormitory
- C = student association of class of 2004
- D = set of students taking discrete mathematics
- E = group of students who ride motorbikes to go to campus
a. Statement
"All STT Telkom students of the 2004 intake who ride motorbikes to go to campus" can be expressed in set operation notation as follows:
(A ∩ C) ∩ E
b. Statement
"All STT Telkom students who live in the dormitory and do not take discrete mathematics" can be expressed in set operation notation as follows:
A ∩ B ∩ D
c. Statement
"all students of the 2004 intake who do not live in dormitories or do not bring motorbikes to go to campus" can be expressed in set operation notation as follows:
C ∩ (B ∪ E)
4. Difference
The difference between two sets is denoted by the symbol '--'. Suppose A and B are sets, then the difference between A and B is denoted by A -- B = { x | x ∈ A and x ∉ B } = A ∩ B
Difference
Example of difference:
If A = { 1, 2, 3, ..., 10 } and B = { 2, 3, 5, 7}, then A -- B = { 1, 4, 6, 8, 9 } and B -- A = ∅
5. Symmetric Difference
The common difference between two sets is denoted by the symbol '⊕'. Suppose A and B are sets, then the common difference between A and B is denoted by:
A ⊕ B = (A ∪ B) -- (A ∩ B)
= (A -- B) ∪ (B -- A)
If expressed in Venn diagram form it is:
Symmetric Difference
Examples of different pairs:
If A = { 2, 3, 5, 7 } and B = { 1, 2, 3, 4, 5 }, then A ⊕ B = { 1, 4, 7 }
The difference between the pairs satisfies the following properties:
A ⊕ B = B ⊕ A (commutative law)
(A ⊕ B ) ⊕ C = A ⊕ (B ⊕ C ) (associative law)
6. Cartesian product
The Cartesian product between two sets is denoted by the '×' symbol. Suppose A and B are sets, then the Cartesian product between A and B is denoted by:
A × B = {(a, b) | a ∈ A and b ∈ B}
Examples of Cartesian multiplication:
- Let C = {1, 2, 3}, and D = { a, b }, then C × D = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }
- Let A = B = the set of all real numbers, then A × B = the set of all points in the plane.
Suppose there are two sets with finite cardinality, then the cardinality of the resultant set of a Cartesian product between the two sets is the product of the cardinalities of each set. Thus, if A and B are finite sets, then:
|A × B| = |A| . |B|
The ordered pair (a, b) is different from (b, a), in other words (a, b) ≠ (b, a). This argument means that the Cartesian product is not commutative, that is,
A × B ≠ B × A
where A or B is not an empty set. If A = ∅ or B = ∅, then
A × B = B × A = ∅
The laws that apply to set operations are as follows:
1. Law of identity:
- A ∪ ∅ = A
- A ∩ U = A
2. Null/dominance law:
- A ∩ ∅ = ∅
- A ∪ U = U
3. Law of complement:
- A ∪ A = U
- A ∩ A = ∅
4. Idempotent law:
- A ∪ A = A
- A ∩ A = A
5. Law of involution:
6. Law of absorption:
- A ∪ (A ∩ B) = A
- A ∩ (A ∪ B) = A
7. Commutative law:
- A ∪ B = B ∪ A
- A ∩ B = B ∩ A
8. Associative law:
- A ∪ (B ∪ C) = (A ∪ B) ∪ C
- A ∩ (B ∩ C) = (A ∩ B) ∩ C
9. Distributive law:
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
10. De Morgan's Law:
11. Law of complement:
