Axioms and Useful Theorems of Boolean Algebra

Question 1

Cardinality

Answer 1

Set B contains at least two elements, a, b, such that a ≠ b

Question 2

Closure

Answer 2

(1) a + b is in B
(2) a * b is in B

Question 3

Commutativity

Answer 3

(1) a + b = b + a
(2) a * b = b * a

Question 4

Associativity

Answer 4

(1) a + (b + c) = (a + b) + c
(2) a * (b * c ) = (a * b) * c

Question 5

Distributivity

Answer 5

(1) a + (b * c) = (a + b) * (a + c)
(2) a * (b + c) = (a * b) + (a * c)

Question 6

Identity

Answer 6

(1) a + 0 = a
(2) a * 1 = a

Question 7

Complementarity

Answer 7

(1) a + a’ = 1
(2) a * a’ = 0

Question 8

Operations with 0 and 1

Answer 8

(1) x + 0 = x
(2) x + 1 = 1
(3) x * 1 = x
(4) x * 0 = 0

Question 9

Idempotent Theorem

Answer 9

(1) x + x = x
(2) x * x = x

Question 10

Involution Theorem

Answer 10

(x’)’ = x

Question 11

Simplification Theorems

Answer 11

(1) x * y + x * y’ = x
(2) (x + y) * (x + y’) = x
(3) x + x * y = x
(4) x * (x + y) = x
(5) (x + y’) * y = x * y
(6) (x * y’) + y = x + y

Question 12

DeMorgan’s Law

Answer 12

(1) (x + y + z + … )’ = x’ * y’ * z’ * …
(2) (x * y * z *…)’ = x’ + y’ + z’ + …

Question 13

Consensus Theorem

Answer 13

(1) xy + yz + x’z = xy + x’z

Question 14

N-Type MOSFET

Answer 14

• Low: DISCONNECTED
• High: CONNECTED

Question 15

P-Type MOSFET

Answer 15

• Low: CONNECTED (closed switch)
• High: DISCONNECTED (open switch)