The principle of duality is a kind of ubiquitous property of the algebraic structure, in which two concepts are interchangeable only if all the results of one formulation equally apply to another. This concept is called double formulation. We will exchange Unions(∪) in Intersections(∩) or Intersections() in Union() and also exchange Universal Set in Zero Set(∅) or Zero Set in universal(U) to get the double instruction. If we exchange the symbol and receive this declaration ourselves, it is called a self-dual declaration. The rules of mathematical logic define methods for the argumentation of mathematical statements. The Greek philosopher Aristotle was the pioneer of logical thought. Logical thinking provides the theoretical basis for many areas of mathematics and thus computer science. It has many practical applications in computer science such as computer machine design, artificial intelligence, definition of data structures for programming languages, etc. The principle of duality of the whole is the strongest and most important property of the menalgebra. He said that the double declaration could be obtained for any true declaration relating to the whole by fixing the union in the intersection and the change from universal (U) to zero.
The opposite of this inclusion is also the case. In set theory, we can exchange the “contains” and “contains in” relationships, where the union becomes the intersection and vice versa. This concept is called auto-dual because in this concept, the original structure remains unchanged. If the instruction is identical to its own dual, it is called an auto-dual. Symbolic logic can be called the simplest type of logic. This 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 the logical expression using symbols and variables, so they can eliminate inaccuracies. The main concern of symbolic logic is the analysis of the accuracy of logical laws such as the hypothetical syllogism, the law of contradiction, etc. Symbolic logic also contains the same self-duality when we exchange “is implicit by” and “implicit” with the connective logics “or” and “and”. So we can say that if we exchange two words, one true statement can be obtained from another. If we perform duality, then the union is replaced by the crossing, or the crossing is replaced by the union.
Different systems have underlying lattice structures: symbolic structure, set theory, and projective geometry. These systems also contain the principles of duality. The principle of the concept of duality should not be avoided or underestimated. It has the ability to provide multiple sets of theorems, concepts, and identities. To explain the principle of duality of sets, suppose that S is any identity that includes complementary sets and operations, a union, an intersection. Suppose we get the S* of S by substituting ∪ → ∩ Φ. In this case, the S* statement is also true, and S* can also be called the dual S statement. As mentioned earlier, it`s called $p rightarrow q$.
(ii) The dual of p ∧ [¬ q ∨ ( p ∧ q) ∨ ¬ r ] is p ∨ [¬ q ∧ ( p ∨ q) ∧ ¬ r ]. Negation ($lnot$) − The negation of a sentence A (written $lnot A$) is false if A is true, and is true if A is false. Example of a conditional instruction – “If you do your homework, you will not be punished.” Here is “you do your homework” the hypothesis, p, and “you will not be punished” is the conclusion, q. The double declaration must be clear, so if we change the language of a statement to specify the instruction, it is clearly understood. The double statement “Two lines determine a point” is clear from the double statement “Two lines intersect at a point”. If we specify a line and think of it as a pencil or set that contains all the lines on which it is located, then the statement “Two dots intersect in one line” is also clear. This concept is also twofold in itself, because in this concept we consider the line as a set of all the points in it. The dual of an instruction formula is obtained by replacing ∨ with ∧, ∧ with ∨, T with F F with T. A dual is obtained by replacing T (tautology) with (contradiction), F and T. If a compound instruction S1 contains only ¬, ∧ and ∨ and the instruction S2 comes from S1 replacing ∧ with ∨, and ∨ with ∧ then S1 is a tautology exactly when S2 is a contradiction. Two statements X and Y are logically equivalent if one of the following two conditions is true – if and only if ($ Leftrightarrow $) – $A Leftrightarrow B$ is a bi-conditioned logical connection that is true if p and q are equal, that is, both are false or both are true.
(1) The symbol ¬ is not changed when the dual is found. Inverse – An inversion of the conditional statement is the negation of the hypothesis and conclusion. If the statement is “If p, then q”, the reverse is “If it is not p, then not q”. Thus, reversing $p rightarrow q$ $ lnot p rightarrow lnot q$. The truth tables in each statement have the same truth values. A compound statement is in conjunctive normal form when obtained by operating AND between variables (including the negation of variables) associated with the ORs. In terms of set operations, it is a composite statement obtained by intersection between variables associated with unions. $(A land B) lor (A land C) lor (B land C land D)$ (3) The special statements T (tautology) and F (contradiction) are duels between them. A sentence is a set of declarative statements that has either a truth value of “true” or a value of truth “false”. A propositional consists of propositional variables and connections.
We designate instruction variables by capital letters (A, B, etc.). The connectors connect the instruction variables. Example − The dual of $(A cap B) cup C$ is $(A cup B) cap C$ Projective geometry contains a lattice structure. This structure can be seen by organizing plans, points, and lines using the inclusion relationship. In the projective geometry of the plane, double statements can be described by exchanging a straight line and a point. The double declaration 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. Implication / if-then $(rightarrow)$ is also known as a conditional statement.
It consists of two parts – All assemblies are replaced by their complement when we remove the dust after the application of c. This means that unions are replaced by crossovers and vice versa. Example − Proof $(A lor B) land lbrack ( lnot A) land (lnot B) rbrack$ is a contradiction Example − Proof $(A lor B) land (lnot A)$ a contingency Here we can see the logical values of $lnot (A lor B) and lbrack (lnot A) land (lnot B) rbrack$ equal, so the statements are equivalent. A tautology is a formula that always applies to each value of its proposition variable.