1. Linear Algebra

2. Circuits The circuits in computers and other input devices have inputs, each of which is either a 0 or 1, the output is also 0s and 1s. Circuits can be built using any basic element that has two different states.

3. Circuits We use Boolean algebra rules to design circuits. – Boolean algebra provides the operations and rules for working with the set {0,1}. The operation of a circuit is defined by a Boolean function that specifies the values of an output for each set of inputs.

4. Boolean functions Complement (denoted by a bar) Sum (+ OR) Product (. AND) → 0 corresponds to F (false) and 1 corresponds to T (true).

5. Example 1. 0 + (0+1) = 0.. How? Translate into a logical equivelance: (T ̂ F) ͮ (T ͮ F)

6. Boolean functions Every Boolean function can be represented using the three Boolean operators 0, +, and -.

7. Logic Gates

8. Gates Boolean algebra is used to model the circuitry of electronic devices. Each input and output is represented by a zero or one. The basic element of circuits are called gates. Each type of gate implements a Boolean operation. →We build circuits using gates.

9. Inverter Gate (not) A Accepts the value of one Boolean variable as input and produces the complement as output.

10. OR Gate A + B

11. AND Gate A. B A B

12. A Circuit Example Circuits can be constructed using a combination of inverters, OR gates, and AND gates. xy + -xy x y x y