B4 Identity Laws XVO x XAI x B5 Complementation Laws x X 0 x

Last updated: 10/12/2023

B4 Identity Laws XVO x XAI x B5 Complementation Laws x X 0 x V x I We write this algebraic structure as B X V A 0 1 or simply B if the context makes the meaning of tlie other terms clear The two operations V and A are called the join operation and meet operation respectively The unary operation is called the complementation From our discussion preceding the definition above you would agree that the set S of propositions is a Booleaii algebra where T and F will do the job of I and O respectively Thus S A V F T is an example of a Boolean algebra We give another example of a Boolean algebra below Example 1 Let X be a non empty set and P X denote its power set i e P X is the set consisting of all the subsets of the set X Show that P X is a Boolean algebra Solution We take the usual set theoretic operations of intersection n union u and complementation in P X as the three required operations Let and X play the roles of O and I respectively Then from MTE 04 you can verify that all the coriditions for P X U n X to be a Boolean algebra hold good For instance the identity laws B4 follow from two set theoretic facts hamely the intersection of any subset with the whole set is the set itself arid the union of any set with the empty set is the set itself On the otlier hand the complementation laws B5 follow from another set of facts from set theory namely the intersection of any subset with its complement is the empty set and the union of any set with its complement is the whole set Yet another example of a Boolean algebra is based on switching circuits For this we first need to elaborate on the functioning of ordinary switches in a mathematical way In fact we will present the basic ides which helped tlic American C E Shannon to detect tlic connection between the functioning of switches and Boole s symbolic logic You may be aware of the functioning of a simple on off switch which is commonly used as an essential component in the electric or electronic networking systems A switch is a clevice which allows the current to flow only when it is placed in tlic ON position i e when the gap is closed by a conducting rod Thus the ON position of a switch is one state of a switch called a closed state The other state of a switch is the open state when it is placed in the OFF position So a switch has two stable states There is another way to talk about the functioning of a switch We can denote a switch by x and use the values 0 and 1 to depict its two states i e to convey that x is open we write x 0 and to convey that x is closed we write x 1 see Fig 2 TXT These values which clcnote the state of a switch x are called the state values s v in short of that switch Boolean Algebra Circ x 1 x 0 Fig 2 OFF ON p tion