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This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it: [6] "One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion.
Because the logical or means a disjunction formula is true when either one or both of its parts are true, it is referred to as an inclusive disjunction. This is in contrast with an exclusive disjunction, which is true when one or the other of the arguments are true, but not both (referred to as exclusive or, or XOR).
For example, [] is the smallest subring of C containing all the integers and ; it consists of all numbers of the form +, where m and n are arbitrary integers. Another example: Z [ 1 / 2 ] {\displaystyle \mathbf {Z} [1/2]} is the subring of Q consisting of all rational numbers whose denominator is a power of 2 .
Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd ...
In statistics, a k-th percentile, also known as percentile score or centile, is a score below which a given percentage k of scores in its frequency distribution falls ("exclusive" definition) or a score at or below which a given percentage falls ("inclusive" definition); i.e. a score in the k-th percentile would be above approximately k% of all scores in its set.
In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object. [1]For example, if m and n are two logical matrices, then
For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval; the set of all positive real numbers is an interval, denoted (0, ∞); the set of all real numbers is an interval, denoted (−∞, ∞); and any single real number a is an interval, denoted [a, a].
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. It is a fallacy of equivocation between the operations ...