1. Dr. Clincy Professor of CS
CS Chapter 3 (3A and )
Dr. Clincy
Professor of CS
Dr. Clincy
Lecture
Slide 1 1

2. Consensus Theorem
F(x,y,z) = xy + x′z + yz

3. Consensus Theorem
Working backwards and adding a term
Dr. Clincy

4. Logic Gates We have looked at Boolean functions in abstract terms.
In this section, we see that Boolean functions are implemented in digital computer circuits called gates.
A gate is an electronic device that produces a result based on two or more input values.
In reality, gates consist of one to six transistors, but digital designers think of them as a single unit.
Integrated circuits contain collections of gates suited to a particular purpose.
Dr. Clincy
Lecture

5. Logic Gates The three simplest gates are the AND, OR, and NOT gates.
They correspond directly to their respective Boolean operations, as you can see by their truth tables.
Dr. Clincy
Lecture

6. Logic Gates Another very useful gate is the exclusive OR (XOR) gate.
The output of the XOR operation is true only when the values of the inputs differ.
Note the special symbol  for the XOR operation.
Dr. Clincy
Lecture

7. Logic Gates
NAND and NOR are two very important gates. Their symbols and truth tables are shown at the right.
Dr. Clincy
Lecture

8. Logic Gates
NAND and NOR are known as universal gates because they are inexpensive to manufacture and any Boolean function can be constructed using only NAND or only NOR gates.
Dr. Clincy
Lecture

9. Logic Gates Gates can have multiple inputs and more than one output.
A second output can be provided for the complement of the operation.
We’ll see more of this later.
Dr. Clincy
Lecture

10. Combinational Circuits
We have designed a circuit that implements the Boolean function:
This circuit is an example of a combinational logic circuit.
Combinational logic circuits produce a specified output (almost) at the instant when input values are applied.
In a later section, we will explore circuits where this is not the case (sequential circuits).
Dr. Clincy
Lecture

11. Combinational Circuits
We have designed a circuit that implements the Boolean function:
This circuit is an example of a combinational logic circuit.
Combinational logic circuits produce a specified output (almost) at the instant when input values are applied.
In a later section, we will explore circuits where this is not the case (sequential circuits).
Dr. Clincy
Lecture

12. Combinational Circuit – Example 1 - XOR
C.How do we construct a truth table for the expression ?
1.Determine AND values
2.Determine OR values
A. The expression for this circuit is this - explain
B.The function is in the “sum-of-products” form – which is really this
Digital logic implies the following order: NOT, AND, OR
D.When comparing the output to the input pattern – we see we have an XOR case – we could replace that circuit with a simple XOR gate

13. Combinational Circuit – Example 2
Explain how to extract from the truth table the expression for the circuit for f1
Two
3-variable
functions
First, figure out the PRODUCTS that make f1 true/high/one – NOT the variables that 0 so that when they are ANDed, the result is 1 x x x f f 1 2 3 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

14. Combinational Circuit – Example 2 continuing

15. Combinational Circuit – Example 2 continuing
Given the expression initially, evaluate the expression and see if it is equal to the original output for f1
Evaluation of
the
expression x x + x x 1 2 2 3 x x x x x x x x x + x x = f 1 2 3 1 2 2 3 1 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

16. Combinational Circuit – Example 2 continuing
Two
3-variable
functions
Lets do f2 x x x f f 1 2 3 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

17. Truth-table technique for proving equivalence of expressions
This algebraic expression represents the distributive identity
We can prove it is correct by constructing a truth table for the left-handside and the right-handside and seeing if they match
Left-hand
side
Righ
t-hand
side w y z y + z w ( y + z ) w y w z w y + w z 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1