1. Timing Diagrams: I did not cover them in class but you are expected to know how to draw one and synthesize a truth table or function from one.

2. Analysis versus Synthesis

1. Analysis: Given an existing function, determine its output based on a given set of inputs. Frequently we produce either a truth table or a timing diagram as a product of analysis.

2. Synthesis: Design a function from a truth table or timing diagram

3. We showed how truth tables can be used to prove the equivalence of two functions. We specifically proved that x1' + x1*x2 = x1' + x2. I am using ' to denote not.

4. Boolean Algebra

1. Axioms: A basic premise--it requires no proof--it is taken as fact. The first three axioms define AND and OR. The fourth axiom defines NOT
2. Single-Variable Theorems: Can be proven through a brute-force application of the axioms
3. 2/3 Variable Theorems: Can be proven via algebraic manipulation using earlier axioms and theorems

5. Duality: The axioms and theorems are presented in pairs to emphasize their duality. Given a logic expression, its dual is obtained by replacing all + operators with * operators, and vice versa, and replacing all 0s with 1s, and vice versa. The dual of any true statement is also a true statement

6. Ways to prove that two functions are equivalent

1. show that their truth tables are identical (i.e., for every possible combination of inputs show that the two functions produce the same output)
2. algebraic manipulation using the axioms and theorems of Boolean algebra
3. Venn diagrams

7. An example of proving two functions equivalent using algebraic manipulation. I am presenting two proofs to show that often there is more than one way to prove a theorem. Proving equivalence and simplifying functions via algebraic manipulation is an art, not a science. In Chapter 4 we will discuss algorithmic means for simplification that can be implemented in software tools.

Prove: x1' + x1*x2 = x1' + x2

Proof 1:
Initial FunctionEquivalent FunctionAxiom/Theorem Used
To Obtain Equivalent Fct
x1'+x1*x2(x1'+x1)*(x1'+x2)12b
1*(x1'+x2)8b
x1'+x26a

Proof 2:
Initial FunctionEquivalent FunctionAxiom/Theorem Used
x1'+x1*x2x1'+x216a