CMSC 250CMSC 250Discrete StructuresDiscrete StructuresLogic Applications(Circuits & Adders)11 June 2007 Logic Applications 2CircuitsCircuits AND gate OR gate NOT gate11 June 2007 Logic Applications 3Combining & Combining & Determining I/O RelationshipDetermining I/O Relationship P ∨∨∨∨ ~ (Q ^ R)11 June 2007 Logic Applications 4Draw the Circuit for:Draw the Circuit for:P, Q & RareinputsSimplifybefore buildingthe circuit00000100001001101001010110111111OutputRQP11 June 2007 Logic Applications 5Number ConversionsNumber Conversions Base of the Number System– 10 (decimal), 2 (binary), 8 (octal), 16 (hexadecimal)– Tells how many different numerals are used– Determines the value of each place Conversions from anything to Base 10– Use the definition of the number system Conversions from Base 10 to anything– Use repeated integer division11 June 2007 Logic Applications 6Addition of Binary NumbersAddition of Binary Numbers Carry if the number would be too large for the number system – if it is greater than 11001+ 111011+ 111011+1111001+ 1011 June 2007 Logic Applications 7Addition of Binary NumbersAddition of Binary Numbers Carry if the number would be too large for the number system (larger than 7 or 15)2658+ 338ABC16+ 1216CDE16+ED167238+ 12811 June 2007 Logic Applications 8Using a Circuit for AdditionUsing a Circuit for Addition Write as a logic expression Translate to circuitsOutputInput0111100110100000SumCarryQP11 June 2007 Logic Applications 9Half AdderHalf AdderQP and Q are binary values (1 bit each)sum = (P v Q) ^ ~(P ^ Q)carry = (P ^ Q)Psumcarry11 June 2007 Logic Applications 10Full AdderFull Adder P, Q and R are binary digits P + Q + R gives sum value and carry valuehalf-adder #1half-adder #2PQRS1C1CSC211 June 2007 Logic Applications 11Parallel AddersParallel Adders Chain these half adders and full adders together for multi-bit addition X1X2X3+ Y1Y2Y3 = CA1A2A3 half-adderfull-adderfull-adderX3Y3X2Y2X1Y1A3carrycarrycarryA2A111 June 2007 Logic Applications 122's Compliment2's ComplimentTo represent negative values using binary1.Find the binary equivalent of the absolute value.2.Pad on the left to completely fill the bits.3.Switch all of the 1's to 0's and 0's to 1's.4.Add 1 to the result.Find the 8-bit 2's compliment representation of -431.43102.10101123.0010101124.1101010025.110101012 =
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