A combinationacombinational circuit has inputs, outputs and an internal logic circuit Constitutes a mapping from the inputs to the outputs A sequential circuit has inputs, outputs, an internal logic circuit and binary cells as memory A binary cell is a memory device with two possible states. | Combinational vs. Sequential Circuits A combinational circuit has inputs, outputs and an internal logic circuit Constitutes a mapping from the inputs to the outputs A sequential circuit has inputs, outputs, an internal logic circuit and binary cells as memory Outputs Binary Cell Logic Circuit A binary cell is a memory device with two possible states. Inputs Boolean Algebra Variables in a Boolean algebra take on values that are not numbers but logical values: TRUE or FALSE x = TRUE, y = FALSE George Boole 1815 - 1864 “It is not of the essence of mathematics to be conversant with the ideas of number and quantity.” ~ An Investigation into the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities Boolean Algebra Provides the basis for digital logic – a logic based on processing discrete (rather than analog) signals Can be used to describe the states of switching circuits Led to the development of digital computers Claude Shannon 1916 - 2001 “It just . | Combinational vs. Sequential Circuits A combinational circuit has inputs, outputs and an internal logic circuit Constitutes a mapping from the inputs to the outputs A sequential circuit has inputs, outputs, an internal logic circuit and binary cells as memory Outputs Binary Cell Logic Circuit A binary cell is a memory device with two possible states. Inputs Boolean Algebra Variables in a Boolean algebra take on values that are not numbers but logical values: TRUE or FALSE x = TRUE, y = FALSE George Boole 1815 - 1864 “It is not of the essence of mathematics to be conversant with the ideas of number and quantity.” ~ An Investigation into the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities Boolean Algebra Provides the basis for digital logic – a logic based on processing discrete (rather than analog) signals Can be used to describe the states of switching circuits Led to the development of digital computers Claude Shannon 1916 - 2001 “It just happened that no one else was familiar with both fields at the same time.” LOW HIGH Logical Variables Logical variables take only two values: Voltage Level [V] In actual computers, these are represented by voltage levels Logical Operators Logical operators mathematically associate one or more (logical) input values with a single (logical) output value Logic gates are the physical devices that perform logical operations (for instance, in circuits) Basic Logical Operators One Input Two (or More) Inputs NOT (negation) OR (disjunction) AND (conjunction) Switching Circuits Boolean algebra was first applied to switching circuits OR gate AND gate Light Bulb Battery Light Bulb Battery x y x y x y y x In the input, use a '1' to represent a closed switch, a '0' for an open switch; in the output, use a '1' for a lit bulb, a '0' for an unlit bulb Truth Tables Since there are only two possible values (0 or 1) for any logical variable, we can define each of the logical operations by .
