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Venn diagram for "A or B", with inclusive or (OR) Venn diagram for "A or B", with exclusive or (XOR). The fallacy lies in concluding that one disjunct must be false because the other disjunct is true; in fact they may both be true because "or" is defined inclusively rather than exclusively.
Although the type of a logical disjunction expression is Boolean in most languages (and thus can only have the value true or false), in some languages (such as Python and JavaScript), the logical disjunction operator returns one of its operands: the first operand if it evaluates to a true value, and the second operand otherwise.
Whereas expressions denote mainly numbers in elementary algebra, in Boolean algebra, they denote the truth values false and true. These values are represented with the bits , 0 and 1. They do not behave like the integers 0 and 1, for which 1 + 1 = 2 , but may be identified with the elements of the two-element field GF(2) , that is, integer ...
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.
A clause is a disjunction of literals (or a single literal). A clause is called a Horn clause if it contains at most one positive literal. A formula is in conjunctive normal form (CNF) if it is a conjunction of clauses (or a single clause). For example, x 1 is a positive literal, ¬x 2 is a negative literal, and x 1 ∨ ¬x 2 is a clause.
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true if its operand is false and a value of false if its operand is true. The truth table for NOT p (also written as ¬p , Np , Fpq , or ~p ) is as follows:
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic.
For example, the logical disjunction function or with boolean values used for true (1) and false (0) is positive unate. Conversely, Exclusive or is non-unate, because the transition from 0 to 1 on input x0 is both positive unate and negative unate, depending on the input value on x1.