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For instance, had been declared as a subset of , with the sets and not necessarily related to each other in any way, then would likely mean instead of . If it is needed then unless indicated otherwise, it should be assumed that X {\displaystyle X} denotes the universe set , which means that all sets that are used in the formula are subsets of X ...
In Boolean algebra, the inclusion relation is defined as ′ = and is the Boolean analogue to the subset relation in set theory. Inclusion is a partial order . The inclusion relation a < b {\displaystyle a<b} can be expressed in many ways:
The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.
Boolean algebra also deals with functions which have their values in the set {0,1}. A sequence of bits is a commonly used example of such a function. Another common example is the totality of subsets of a set E: to a subset F of E, one can define the indicator function that takes the value 1 on F, and 0 outside F.
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854.
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".
In this poset, 60 is an upper bound (though not a least upper bound) of the subset {,,,}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a ...
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, " is less than " is a relation on the set of natural numbers ; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3 ), and likewise between 3 and 4 (denoted as 3 < 4 ), but not between the ...