Boolean simplification(1.4.3 c)

Question 1

Double negation

Answer 1

¬(¬A) = A

Question 2

Commutation

Answer 2

the order of variables for an operator doesn’t matter
A∧B = B∧A
A∨B = B∨A

Question 3

Association

Answer 3

allows us to fiddle with/remove brackets
OR
A∨(B∨C) = (A∨B)∨C = A∨B∨C
AND
A∧(B∧C) = (A∧B)∧C = A∧B∧C

Question 4

Distribution

Answer 4

OR
A∧(B∨C) = (A∧B)∨(A∧C)
AND
A∨(B∧C) = (A∨B)∧(A∨C)

Question 5

Absorption

Answer 5

applies when a variable is referenced inside and outside a bracket
OR
A∨(A∧B) = A
A∧(A∨B) = A

Question 6

De Morgans first law

Answer 6

¬(A∨B) = ¬A∧¬B
the operator changes

Question 7

De Morgans second law

Answer 7

¬(A∧B) = ¬A∨¬B
the operator changes

Question 8

AND gate rules

Answer 8

if an input is zero the input will always output zero(X∧0 = 0)
if an input is always one the output will will always be the other input(X∧1 = X)
if x is zero than both inputs are zero if x is one both inputs are one (X∧X = X)
if an input is one then another must be zero and vice versa(X∧¬X = 0)

Question 9

OR gate rules

Answer 9

if an input is zero the output will be what the other input is(X∨0 = X)
if an input is always one the output will always be one(X∨1 = 1)
if x is zero then both inputs are zero if x is one then both inputs are one(X∨X = X)
if an input is one then the other leg must be zero and vice versa(X∨¬X = 1)

Question 10

X∧0 = 0 AND gate

Answer 10

if an input is zero the input will always output zero

Question 11

X∧X = X AND gate

Answer 11

if x is zero than both inputs are zero if x is one both inputs are one

Question 11

X∧1 = X AND gate

Answer 11

if an input is always one the output will will always be the other input

Question 12

X∧¬X = 0 AND gate

Answer 12

if an input is one then another must be zero and vice versa

Question 13

X∨0 = X OR gate

Answer 13

if an input is zero the output will be what the other input is

Question 14

X∨1 = 1 OR gate

Answer 14

if an input is always one the output will always be one

Question 15

X∨X = X OR gate

Answer 15

if x is zero then both inputs are zero if x is one then both inputs are one

Question 16

X∨¬X = 1 OR gate

Answer 16

if an input is one then the other leg must be zero and vice versa