this definition Now using Tables 1 and 2 you can check that the five laws B1 B5 hold good Thus S par ser 1 0 1 is a Boolean algebra A Boolean algebra whose underlying set has only two elements is very important in the study of circuits We call such an algebra a two element Boolean algebra Throughout the unit we denote this algebra by B From this Boolean algebra we can build many more as in the following example Example 3 Let B BxBx x B e e2 en each e 0 or 1 for n 2 1 be the Cartesian product of n copies of B For ik jk 0 1 1 k n define 11 12 11 12 1 2 Jn i1Aj1 12 A 12 in jn Vi1 j2 11 12 in El a Then B is a Boolean algebra for all n 1 Solution Firstly observe that the case n 1 is the Boolean algebra B Now let us write 0 0 0 0 and I 1 1 1 for the two clcments of B consisting of n tuples of 0 s and 1 s respectively Using the fact that B is a Boolean algebra you can check that B3 with operations as clcfined above is a Boolean algebra for every n 2 1 in Vj1 12 Vj2 in Vin and The Boolean algebras B 121 called switching algebras are very useful for the study of the hardware and software of digital computers We shall now state without proof some other properties of Boolean algebras which can be deduced from the five laws B1 B5 Theorem 1 Lot B S V A O I be a Boolean algebra Then the following laws hold Vx y ES x V x x x Ax x a Idempotent laws b Absorption laws c Involution law d De Morgan s laws x V x Ay x x A XV y x x x xVy x Ay x Ay x Vy In fact you have already come across some of these properties for the Boolean algebra S of propositions in Unit 1 In the following exercise we ask you to verify them Verify the identity laws and absorption laws for the Boolean algebra S A V T F of propositions b Verify the absorption laws for the Boolean algebra P X U n X 1 In Theorem 1 you may have noticed that for each statement involving V and A there is an analogous statement with A instead of V and V instead of A This is not a coincidence as the following definition and result shows Definition Ifp is a proposition involving A and V the dual of P denoted by pd is the proposition obtained by replacing each occurrence of A and or V in p by V and or A respectively in p For example x V x A y x is the dual of x A x V y x Boolean A