In this paper, the authors proposed two full adder circuits such as mixed Shannon and Shannon theorem based adder circuits. The mixed adder circuit developed by using MCIT (Multiplexing Control Input Technique) for the sum operation and Shannon technique for carry and other one designed completely by using Shannon theorem. The ALU (Arithmetic Logic Unit) circuit consists of AND, OR, multiplexer and adder circuits that are designed by using proposed Shannon theorem. The 32-bit ALU circuits are analyses by BSIM 4 parameter analyzer. The power dissipation of total circuit, propagation delay and area are analyzed for 32 bit ALU.