- 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.
- Analysis versus Synthesis
- 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.
- Synthesis: Design a function from a truth table or timing diagram

- 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.
- 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**. - Boolean Algebra
- 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
- Single-Variable Theorems: Can be proven through a brute-force application of the axioms
- 2/3 Variable Theorems: Can be proven via algebraic manipulation using earlier axioms and theorems

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

- 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 Function Equivalent Function Axiom/Theorem Used

To Obtain Equivalent Fctx1'+x1*x2 (x1'+x1)*(x1'+x2) 12b 1*(x1'+x2) 8b x1'+x2 6a Proof 2:

Initial Function Equivalent Function Axiom/Theorem Used

x1'+x1*x2 x1'+x2 16a