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There is an inclusion–exclusion principle for finite multisets (similar to the one for sets), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an odd number of the given multisets, while in the second sum we consider all possible ...
The board B is any subset of the squares of a rectangular board with n rows and m columns; we think of it as the squares in which one is allowed to put a rook. The coefficient , r k ( B ) of x k in the rook polynomial R B ( x ) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B .
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
One common convention is to associate intersection = {: ()} with logical conjunction (and) and associate union = {: ()} with logical disjunction (or), and then transfer the precedence of these logical operators (where has precedence over ) to these set operators, thereby giving precedence over .
One may define the operations of the algebra of sets: union(S,T): returns the union of sets S and T. intersection(S,T): returns the intersection of sets S and T. difference(S,T): returns the difference of sets S and T. subset(S,T): a predicate that tests whether the set S is a subset of set T.
4.3 Permutations of multisets. 4.4 ... A permutation matrix is an n × n matrix that has exactly one entry 1 in each column and in ... then it must be the union of k ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
The chart below is similar to the chart above, but instead of showing the formulas, it gives an intuitive understanding of their meaning using the familiar balls and boxes example. The rows represent the distinctness of the balls and boxes. The columns represent if multi-packs (more than one ball in one box), or empty boxes are allowed.