- I am using Carpinelli chapter 2.
- We are looking at section 2 of the outline.
- 2 a : Combinational logic is a circuit where the output is completely determined by the input.
- Sequential logic, is a circuit which has a state and the output is determined by a combination of the input and that state.
- This is a circuit with a memory.

- Everything we have looked at so far is combinational.
- We will look at each of these in the future.

- We have covered 2b and 2c
- Another warning:
- Truth tables will be built with 0 and 1 not T/F
- Rows in truth tables will be ordered from 0 to $2^n-1$ for n inputs.
- You will use the logical operators +,· and . If you need ' will do for not.

- Boolean Properties.
- Closure: all of the boolean operators will produce a value of either 0 or 1.
- This is like addition under the integers, add any two integers and you get another integer.
- Division is not closed under the integers. 1/2 is not an integer.

- There exists an identity element
- In arithmetic:
- a + 0 = a
- 0 is the additive identity
- a × 1 = a
- 1 is the multiplicative identity.

- What is the identity for or? 0
- 0 + 0 = 0
- 1 + 0 = 1

- What is the identity for and? 1
- 0 · 1 = 0
- 1 · 1 = 1

- In arithmetic:
- There is an inverse
- the inverse of a is a
- a · a = 0
- a + a = 1

- Commutative property
- ab = ba, a+b = b+a
a b ab ba a+b b+a 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1

- ab = ba, a+b = b+a
- Associative properties
- (a+b)+c = a+(b+c)
a b c a+b (a+b)+c b+c a+(b+c) 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 - (ab)c = a(bc)
- Table is similar

- (a+b)+c = a+(b+c)
- Distributive properties
- a(b+c) = ab+ac
a b c b+c a(b+c) ab ac ab+ac 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1

- a(b+c) = ab+ac

- Closure: all of the boolean operators will produce a value of either 0 or 1.
- The following are helpful when simplifying circuits
- Idempotence
- When a function takes the same input twice, it is the value of the input.
- This is helpful when simplifying circuits.
a aa a+a 0 0 0 1 1 1

- Involution
- The inverse of the inverse is the item
a a a̿ 0 1 0 1 0 1 - I used a unicode trick for the double bar.
- a̿ is a followed by the unicode character 831 (in decimal)

- The inverse of the inverse is the item
- Absorption
- a(a+b) = a, a+(ab) = a
a b a+b a(a+b) ab a+(ab) 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1

- a(a+b) = a, a+(ab) = a

- Idempotence
- De Morgan's
- a+b = a·b
- ab = a+b