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Logic Gates Claude Shannon Circuits Boolean algebra Overview of ...

Logic Gates Claude Shannon Circuits Boolean algebra Overview of ...

9/10/13 Basic logic

9/10/13 Basic logic gates Combinational circuit rules • Not • And • Or • Nand • Nor • Xor x x y x y x y x y x y x xy x y xyz x+y z x y z x+y+z xy x+y xÅ y 7 • In general, gates can be combined into combinational circuits • Rules: – Never combine two input wires – A single input wire can be split partway and used as input for two separate gates – An output wire can be used as input – No output of a gate can eventually feed back into that same gate 8 Converting between circuits and Boolean expressions • Find the output of the following circuit Converting between circuits and equations • Find the output of the following circuit x y y x+y y (x+y)y x y x y x y x y __ • Answer: (x+y)y – Or (x∨y)∧~y 9 ___ _ _ • Answer: xy – Or ~(~x∧~y) ≡ x∨y 10 Converting between circuits and equations • Write the circuits for the following Boolean algebraic expressions __ a) x+y Converting between circuits and equations • Write the circuits for the following Boolean algebraic expressions _______ b) (x+y)x x y x x+y x y x+y x+y (x+y)x 11 12 2

9/10/13 Writing xor using and/or/not • p ⊕ q ≡ (p ∨ q) ∧ ~(p ∧ q) • x ⊕ y ≡ (x + y)(xy) x y ____ x+y xy xy x y x⊕y 1 1 0 1 0 1 0 1 1 0 0 0 (x+y)(xy) 13 14 OR gate Integrated Circuits • Very advanced (and miniaturized) electronic circuits • Least expensive way to make logic gates in large volumes • Mainly consist of semiconductor devices – Transistors – on/off/amplify • Integrate a large number of transistors onto a tiny chip (die) … 15 16 The 7400 chip, containing four NANDs. Eniac: the first computer 17 18 3

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