De Morgan’s Laws for Logic

Then, quantifying dualities can be extended to modal logic by connecting the operators box (“necessarily”) and Diamond (“possibly”): De Morgan`s laws describe how mathematical statements and concepts are connected by their opposites. In set theory, De Morgan`s laws refer to the intersection and union of sets by complements. In the logic of the proposition, De Morgan`s laws refer to conjunctions and disjunctions of propositions by negation. De Morgan`s laws are also applicable in computer engineering for the development of logic gates. We can also use Morgan`s laws to simplify the denial of $Pimplies Q$: $$eqalign{lnot (Pimplies Q) & iff lnot (lnot Plor Q)cr & iff (lnotlnot P)land (lnot Q)cr & iff Pland lnot Qcr} $$, so that the rejection of $Pimplies Q$ is $Pland lnot Q$. In other words, it`s not that $$P$ implies $$Q$ if and only if $$P is true and $$Q is false. Of course, this is consistent with the truth chart for $P implies Q$, which we have already seen. To relate these quantifying dualities to Morgan`s laws, set up a model with a small number of elements in its field D, such as: Here`s another way to imagine the quantifying versions of De Morgan`s laws. The $forall x,P(x)$ statement is very similar to a large conjunction. For example, if the universe of discourse consists of positive integers, then this is equivalent to the statement that $$P(1)land P(2)land P(3)land cdots $$ or, more succinctly, we could write $$bigwedge_{xin U} P(x), $$ with a notation similar to the “sigma notation” for sums. Of course, this is not really a “statement” in our official mathematical logic, because we do not allow infinitely long formulas.

Similarly, $exists x,P(x)$ can be thought of as $$bigvee_{xin U} P(x). $$ Now, the first quantifier law can be written $$lnotbigwedge_{xin U} P(x) iff bigvee_{xin U} (lnot P(x)), $$, which closely resembles the $$lnot (Pland Q)iff (lnot Plor lnot Q), $$, but with infinite conjunction and disjunction. Note that we also use De Morgan`s laws for $land$ and $lor$ as $$eqalign{ lnot bigwedge_{i=1}^2 (P_i(x)) &iff bigvee_{i=1}^2 (lnot P_i(x))cr lnot bigvee_{i=1}^2 (P_i(x)) &iff bigwedge_{i=1}^2 (lnot P_i(x)).cr} $$ This is heavier, but reflects the close relationship with the quantifying forms of De Morgan`s laws. In his application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to quantization can be understood by constructing models using Kripke semantics. The first of DeMorgan`s laws is verified by the following table. You will be asked to check the second one in an exercise. DeMorgan`s laws are actually very natural and intuitive. Consider the statement , which we can interpret in such a way that it is not true that P and Q are true.

If it is not true that P and Q are true, then at least one of P or Q is false, in which case is true. Thus, means the same as . Logic includes the formulation of laws and De Morgan`s work that lead to the development of relationship theory and the rise of modern symbolic or mathematical logic.n According to De Morgan`s theorem, a NAND gate is equivalent to an OR gate with inverted inputs. Similarly, a NOR door is equivalent to an AND door with inverted entrances. Figure 2.19 shows these Equivalent De Morgan doors for NAND and NOR doors. The two icons that appear for each function are called doubles. They are logically equivalent and can be used interchangeably. De Morgan`s sentences can also be used to express logical expressions that originally did not contain inversion terms in any other way. This, in turn, can be useful for simplifying Boolean equations.

In this use, care must be taken to ensure that the final inversion does not forget, which can easily be avoided by completing both sides of the expression to be simplified before applying De Morgan`s sentence and completing it again after simplification. The following example illustrates this point. De Morgan`s laws follow a similar structure for logical statements. The language of these concepts may seem intimidating, but the concepts themselves are quite simple. In binary images, our erosion approximation ε ̃g and our expansion approximation δ ̃g produce the same results as conventional erosion or expansion. However, the proposed work does not use binary images. Therefore, adjustments are needed so that our morphological approximations proposed on the grayscale images can act as mammograms, accompanied by minimal distortion compared to the morphological operations described in the literature. De Morgan`s laws can be used to simplify the negations of the “one” form and the “all” form; The negations themselves have the same forms, but “reversed”, that is, the negation of a “whole” form is a “certain” form and vice versa. For example, suppose $P(x)$ and $Q(x)$ are formulas. We then have $$lnot forall x (P(x)implicit Q(x)) iff exists x (P(x) land lnot Q(x)) $$ $$lnot exists x (P(x)land Q(x)) iff forall x (P(x)implies lnot Q(x)) $$ The rejection of the phrase “all lawn mowers run on gasoline” is the phrase “any lawn mower does not run on gasoline” (not “no gasoline-powered lawn mowers, “ on the contrary).

We check the first statement and leave the second to an exercise: $$lnot forall x (P(x)implies Q(x)) iff exists x lnot(P(x)implies Q(x)) iff exists x (P(x) land lnot Q(x)) $$ Then the operations of maximum and minimum set theory can be implemented by arithmetic operators approximately. Therefore, it is possible to create morphological approximations based on set terms by arithmetic operators.