Question

1. Let B-(0, 1). Define x + y max(x, y) and x . y-min(x, y), and let the complement of x of be 1-x (ordinary subtraction). Show whether or not B forms a Boolean algebra under these operations. 2. Let S-(0,1 R, and T = { y : 2 < y < 12). Find a one to one correspondence (the actual function) between S and T showing they have the same cardinality. (hint: look at straight lines in the xy-plane)

Answer 1

1.We can easily verify that all axioms commutativity associativity, ab- sorption laws, identity law, distributivity axioms are satisfied by the given definition of + and - Hence 0,2) forms a Boolean algebra 2.Here S(0,1) and T-(2,12). Consider the function f: S-T defined by f(x) 210:r Note that f(xı)-f(x2) ->210r10x2xi- x2 . Hence f is one-one For each ce (2,12) CE (0,1) (c-2) E (0,10) 10 Hence for each c E (2.12) ,ョ 20 2+c-2- c which proves that f is onto Thus f is a bijection from S to T Hence S and T have the same cardinality (0,1) such that f(xo) - 10