## Principle of Duality in Discrete MathematicsThe 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(
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 ∪ ((B ^{C} ∪ A) ∩ B)^{C} = UWhen 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 ∩ ((B ^{C} ∩ A) ∪ B)^{C} = Φ
(A ∪ B ^{C})^{C} ∩ B = A^{C} ∩ BWhen we perform duality, then the union will be replaced by intersection, or intersection will be replaced by the union. (A ∩ B ^{C})^{C} ∪ B = A^{C} ∪ BVarious systems have underlying lattice structures: symbolic structure, set theory, and projective geometry. These systems also contain the principles of duality. ## Projective GeometryA lattice structure is contained in the 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. ## Set TheoryThe 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
In this example, we will use the complement operator to equality of sets, which contains intersections and unions. (A∩B) ^{C}=A^{C}∪B^{C}(A∩B)^{C}=A^{C}∪B^{C} and (A∪B)^{C}=A^{C}∩B^{C}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}=A^{C}∩B^{C}(A∪B)^{C}=A^{C}∩B^{C} and (A∩B)^{C}=A^{C}∪B^{C}## Symbolic Logic
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 |