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De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
Two such polynomials are equal only if the corresponding coefficients are equal. [6] In contrast, two polynomial functions in a variable may be equal or not at a particular value of . For example, the functions = +, = + are equal when = and not equal otherwise. But the two polynomials +, +
Since is a sub -algebra of , the function : is usually not -measurable, thus the existence of the integrals of the form |, where and | is the restriction of to , cannot be stated in general. However, the local averages ∫ H X d P {\textstyle \int _{H}X\,dP} can be recovered in ( Ω , H , P | H ) {\displaystyle (\Omega ,{\mathcal {H}},P ...
The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: not (A or B) = (not A) and (not B) not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning at least one of A or B rather than an "exclusive or" that means exactly one
The same is true for not less than, . The notation a ≠ b means that a is not equal to b; this inequation sometimes is considered a form of strict inequality. [4] It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.
Double negation elimination and double negation introduction are two valid rules of replacement. They are the inferences that, if not not-A is true, then A is true, and its converse, that, if A is true, then not not-A is true, respectively. The rule allows one to introduce or eliminate a negation from a formal proof.
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. The truth table of p EQ q (also written as p = q , p ↔ q , Epq , p ≡ q , or p == q ) is as follows:
Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤ ~ v and v ≤ ~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the