# 14:332:231 Lecture Notes - Lecture 3: Logic Gate, Summation, Logic Optimization

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18 Jul 2018

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CHAPTER FOUR: COMBINATIONAL LOGIC DESIGN PRINCIPLES

4.1 Switching Algebra

● Use positive-logic convention, where 0 is LOW and 1 is HIGH

● 3 Basic Logic Gates

● Axioms

● Theorems

● Theorems of Several Variables

● Precedence: multiplication takes precedence over addition

● Theorems Explained

○ Covering Theorem (T9)

■ We only need to include X in the expression

■ X is said to cover

X*Y

○ Combining Theorem (T10)

■ If the subexpression X*Y + X*Y’ appears in an expression, we can then

replace it with just X

○ Consensus Theorem (T11)

■ The Y*Z term is called the consensus

of X*Y and X’*Z

■ If Y*Z is 1, then either X*Y or X’*Z must also be one

■ Therefore, term Y*Z is redundant and can be dropped

● Theorems of Many Variables Explained

○ Most of these theorems can be proved using a two-step method called finite

induction

■ First proving the theorem is true for n=2

■ Then proving the theorem is true for n=i and n=i+1 (induction step

)

○DeMorgan’s Theorem

■ Examples