1. Section 10.3
Logic Gates

2. Boolean algebra & circuits
4/17/2017
Boolean algebra & circuits
Boolean algebra is used to model electronic circuitry
Each input & output of electronic device can be thought of as a member of the set {0,1}
Electronic devices made up of circuits
Each circuit can be designed using rules of Boolean algebra
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3. Logic gates Basic elements of circuits are called gates
Each type of gate implements a Boolean operation (e.g. Boolean product, Boolean sum, complement)
Combinational circuits or gating networks:
give output that depends only on input, not current state of circuit
have no memory capability

4. Elements of combinational circuits
Inverter: accepts value of a Boolean variable as input and produces the complement as output
OR gate: inputs are values of 2 or more Boolean variables; output is Boolean sum of their values

5. Elements of combinational circuits
AND gate: inputs are values of 2 or more Boolean variables; output is Boolean product of their values
AND gates and OR gates may have multiple inputs

6. Combinations of gates
Combinational circuits can be constructed using combinations of inverters, OR gates, and AND gates
In such circuit combinations, some gates may share inputs
Output from a gate may be used as input by another element

7. Example 1
Construct a circuit to produce the output:

8. Example 2
Construct a circuit to produce the output:

9. Example 3
We can use circuits to model various types of computational operations
For example, suppose a committee of 3 people decides issues for an organization; each one votes yes or no for each proposal that arises, and a proposal passes if it receives at least 2 yes votes

10. Example 3
Let x, y and z represent the voters’ decisions - a value of 1 means a yes vote, and 0 means a no vote
We can design a circuit to output 1 when two or more of x, y and z are 1 by representing the Boolean function xy + xz + yz

11. Example 3
The resulting circuit looks like this:

12. Example 4 Light fixtures may be controlled by more than one switch
Circuits for such fixtures must be designed so that flipping any switch in the circuit will turn the light on when it is off and off when it is on
We will look at such a circuit with 2 switches

13. Example 4 We represent the two switches as Boolean variables x and y
When a switch is open, its value is 0, and 1 when it is closed
We need to define a function F(x,y)=1 when the light is on, 0 when it is off

14. Example 4
Suppose the light is on if both switches are closed; then F(1,1) = 1
Opening either switch turns the light off, so F(0,1) = F(1,0) = 0
If one switch is open (and the light is off), opening the other switch will turn the light on, so F(0,0) = 1
We can get these results if F(x,y) = xy +

15. Example 4
A three-way circuit can be designed in a similar fashion; for the three switches x,y,z the function F(x,y,z) =

16. Adders
Logic circuits can be used to carry out addition of two positive integers from their binary expansions
Such circuits form the basis of computer arithmetic
Half adder: a circuit that takes 2 bits as input, outputting a sum bit and a carry bit; a half adder is a multiple output circuit
A full adder computes the sum and carry bits when 2 bits and a carry are added

17. Half adder Table below shows the possible inputs and corresponding
outputs of a half adder: x and y are the input bits, s is their
sum, and c is the carry:
x y s c
From the table we can see that c = xy
and s = or

18. Half adder

19. Full adder
Inputs to full adder are the bits x and y and the carry ci; outputs
are the sum bit s and the new carry ci+1
x y ci s ci+1
The outputs of the full adder, which are
the sum bit and the new carry, are given
by the sum-of-products expansions:
Sum:
Carry:

20. Full adder
A full adder can be built using half adders to produce the desired output:

21. 4/17/2017
Section 10.3
Logic Gates
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