The principle of duality states that two different concepts can be interchanged and still give the correct answer.
Examples of the principle of duality:
- US => car steering wheel on the front left
- Indonesia => car steering wheel on the front right
Illustration of the Principle of Duality
Regulation:
(a) in the United States,
- cars must drive on the right side of the road,
- on multi-lane roads, the left lane is for overtaking,
- When the red light is on, cars may turn right immediately
(b) in Indonesia,
- cars must drive on the left side of the road,
- on multi-lane lanes, the right lane is for overtaking,
- When the red light is on, cars may turn left immediately
1. The principle of duality in the above case is:
The concepts of left and right are interchangeable in both countries so that the rules that apply in the United States also apply in the United Kingdom.
(Principle of Duality on Sets). Suppose S is an identity involving sets and operations such as ∪, ∩, and complement. If S* is an identity that is the dual of S then by replacing ∪ → ∩, ∩ → ∪, ∅ → U, U → ∅, while the complement is left as is, then these operations on the identity S* are also true.
2. Duality Table of Set Algebra Laws
Examples of the principle of duality:
Suppose A ∈ U where A = (A ∩ B) ∪ (A ∩ B), then on the dual, say U*, the following holds:
A = (A ∪ B) ∩ (A ∪ B)In proving the truth of a statement or representing a statement in another way using the help of sets, there are several ways, including:
A. Proof using Venn diagrams
Example of proof using Venn diagram
Suppose A, B, and C are sets. Show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) using a Venn diagram.
Answer:
This method is carried out not in a formal proof, by describing a number of known sets and shading each desired operation in stages, so that the overall set of results of the operation is obtained.
Both Venn diagrams give the same shaded area. It is evident that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
B. Proof using set algebra
Example 1:
Let A and B be sets. Show that A ∪ (B -- A) = A ∪ B
Answer 1:
A ∪ (B -- A) = A ∪ (B ∩ Ā) (Definisi operasi selisih)
= (A ∪ B) ∩ (A ∪ Ā) (Hukum distributif)
= (A ∪ B) ∩ U (Hukum komplemen)
= A ∪ B (Hukum identitas)Example 2:
Show that for any sets A and B, it holds;
a. A ∪ (A ∩ B) = A ∪ B and
b. A ∩ (A ∪ B) = A ∩ B
Answer 2:
a. A ∪ (A ∩ B) = ( A ∪ A) ∩ (A ∪ B) (H. distributif)
= U ∩ (A ∪ B) (H. komplemen)
= A ∪ B (H. identitas)(b) is the dual of (a)
b. A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B) (H. distributif)
= ∅ ∪ (A ∩ B) (H. komplemen)
= A ∩ B (H. identitas)