Digital Fundamentals

Symbol: A · B or A ∧ B or AB

A  B | Output
0  0 |   0
0  1 |   0
1  0 |   0
1  1 |   1

Output is 1 only when ALL inputs are 1

Symbol: A + B or A ∨ B

A  B | Output
0  0 |   0
0  1 |   1
1  0 |   1
1  1 |   1

Output is 1 when ANY input is 1

Symbol: A' or Ā or ¬A

A | Output
0 |   1
1 |   0

Flips the input

Symbol: (A · B)'

A  B | Output
0  0 |   1
0  1 |   1
1  0 |   1
1  1 |   0

Universal: Can build any gate using only NAND

Symbol: (A + B)'

A  B | Output
0  0 |   1
0  1 |   0
1  0 |   0
1  1 |   0

Universal: Can build any gate using only NOR

Symbol: A ⊕ B

A  B | Output
0  0 |   0
0  1 |   1
1  0 |   1
1  1 |   0

Output is 1 when inputs are DIFFERENT

Symbol: (A ⊕ B)'

A  B | Output
0  0 |   1
0  1 |   0
1  0 |   0
1  1 |   1

Output is 1 when inputs are SAME

Identity:     A + 0 = A       A · 1 = A
Null:         A + 1 = 1       A · 0 = 0
Idempotent:   A + A = A       A · A = A
Complement:   A + A' = 1      A · A' = 0
Double Neg:   (A')' = A

(A + B)' = A' · B'
(A · B)' = A' + B'

"Break the bar, change the sign"

Commutative:   A + B = B + A       A · B = B · A
Associative:   (A+B)+C = A+(B+C)   (AB)C = A(BC)
Distributive:  A(B+C) = AB + AC
               A+BC = (A+B)(A+C)

Absorption:    A + AB = A
               A(A + B) = A

Simplify: AB + AB'

= A(B + B')     [Factor out A]
= A(1)          [Complement law]
= A             [Identity law]

Simplify: (A + B)(A + B')

= A + BB'       [Distributive]
= A + 0         [Complement]
= A             [Identity]