Principle of Duality in Discrete Mathematics
The principle of duality is a type of pervasive property of algebraic structure in which two concepts are interchangeable only if all results held in one formulation also hold in another. This concept is known as dual formulation. We will interchange unions(∪) into intersections(∩) or intersections() into the union() and also interchange universal set into the null set(∅) or null set into universal(U) to get the dual statement. If we interchange the symbol and get this statement itself, it will be known as the self-dual statement.
The dual of (X ∩ Y) ∪ Z is (X ∪ Y) ∩ Z
Duality can also be described as a property that belongs to the branch of algebra. This theory can be called lattice theory. This theory has the ability to involve order and structure, which are common to different mathematical systems. If the mathematical system has the order in a specified way, this structure will be known as lattice.
The principle of duality concept should not be avoided or underestimated. It has the ability to provide several sets of theorems, concepts, and identities. To explain the duality principle of sets, we will assume S be any identity that involves sets, and operation complement, union, intersection. Suppose we obtain the S* from S with the help of substituting ∪ → ∩ Φ. In this case, the statement S* will also be true, and S* can also be known as dual statement S.
Examples of Duality:
A ∪ (B ∩ A) = A
When we perform duality, then the union will be replaced by intersection, or the intersection will be replaced by the union.
A ∩ (B ∪ A) = A
A ∪ ((BC ∪ A) ∩ B)C = U
When we perform duality, then the union will be replaced by intersection, or intersection will be replaced by the union. The universal will also be replaced by null, or null will be replaced by universal.
A ∩ ((BC ∩ A) ∪ B)C = Φ
(A ∪ BC)C ∩ B = AC ∩ B
When we perform duality, then the union will be replaced by intersection, or intersection will be replaced by the union.
(A ∩ BC)C ∪ B = AC ∪ B
Various systems have underlying lattice structures: symbolic structure, set theory, and projective geometry. These systems also contain the principles of duality.
A lattice structure is contained in the projective geometry. This structure can be seen by ordering planes, points, and lines with the help of inclusion relation. In the projective geometry of the plane, the dual statements can be described by interchanging the line and point. The dual statement of projective geometry is "A line can be determined by two points" and "a point can be determined by two lines". In projective geometry, this last statement is always true because parallel lines are not allowed by axioms, but it is sometimes false in Euclidean geometry.
The dual statement must be clear, so when we modify the language of a statement to specify the statement, it will be clearly understood. The dual statement "Two lines determine a point" is clear as compare to the dual statement "Two lines intersect in a point". If we specify a line and consider it as a pencil or set, which contains all lines on which it lies, then the statement "Two points intersect in a line" will also be clear. This concept is also itself dual because, in this concept, we consider line as a set of all points that lie on it.
The principle of duality for the set is the strongest and important property of set algebra. It said that the dual statement could be obtained for any true statement related to set by interchanging union into the intersection and interchanging universal(U) into null. The reverse of this inclusion is also true. In the set theory, we can interchange the relations "contains" and "contained in" with union becoming the intersection and vice-versa. This concept is known as self-dual because, in this concept, the original structure will remain unchanged. If the statement is the same as its own dual, it will be known as self-dual.
In this example, we will use the complement operator to equality of sets, which contains intersections and unions.
(A∩B)C=AC∪BC(A∩B)C=AC∪BC and (A∪B)C=AC∩BC
All the sets will be replaced by their complement when we clear the dust after applying c. That means unions will be replaced by intersections and vice versa.
(A∪B)C=AC∩BC(A∪B)C=AC∩BC and (A∩B)C=AC∪BC
Symbolic logic can be known as the simplest kind of logic. It can save a lot of time in argumentation. Various logical confusions can also be solved by this logic. It also has the ability to represent logical expression with the help of symbols and variables so that they can remove vagueness. The main concern of symbolic logic is the analysis of correctness of logical laws like hypothetical syllogism, contradiction law, etc. Symbolic logic also contains the same self-duality if we interchange "is implied by" and "implied" with the logical connectives "or" and "and". So we can say that if we interchange two words, one true statement can be obtained from another.
p ∪ ((q ∪ p) ∩ q) = 1
When we perform duality, all the symbols will be replaced by their complements. That means unions will be replaced by intersections and vice versa.
p ∩ ((q ∩ p) ∪ q) = 0