Please forward this laws of algebra of sets pdf screen to 158. This article is about algebraic properties of set operations in general.
It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The algebra of sets is the set-theoretic analogue of the algebra of numbers. It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. Several of these identities or “laws” have well established names. The union and intersection of sets may be seen as analogous to the addition and multiplication of numbers. However, unlike addition and multiplication, union also distributes over intersection.
The preceding five pairs of laws—the commutative, associative, distributive, identity and complement laws—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them. The following proposition states six more important laws of set algebra, involving unions and intersections. As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above. As an illustration, a proof is given below for the idempotent law for union. The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection. The following proposition states five more important laws of set algebra, involving complements. Notice that the double complement law is self-dual.
The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws. The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous. Oxford University Press US, 1996. This page was last edited on 10 October 2017, at 14:02. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.
In circuit engineering settings today, there is little need to consider other Boolean algebras, thus “switching algebra” and “Boolean algebra” are often used interchangeably. As with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows. These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.