2.1, 2.2 & 2.3

Question 1

What is Boolean Algebra?

Answer 1

Mathematical model for digital logic circuits
Binary, {1, 0}, {true, false}
It’s a mathematical system

Question 2

Mathematical System definition

Answer 2

a set of elements and one or more binary operations to connect these elements.
Elements: {0,1}
Operations: {・,+ , ‘}
Operations: {and, or, not}

Question 3

Mathematical system properties:
Closure
______
Inverse
______
Associative
_______

Answer 3

Identity
Commutative
Distributive

Question 4

property:
Assumed to be _____
Obvious

Answer 4

true

Question 5

law:
Proven true by ______ and other proven laws

Answer 5

properties

Question 6

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

Answer 6

Associative property

Question 7

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

Answer 7

commutative property

Question 8

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

Answer 8

distributive property

Question 9

A ∧ True = A
B ∨ False = B

Answer 9

identity law

Question 10

A ∨ A’ = True
B ∧ B’ = False

Answer 10

complement law

Question 11

A ∨ A = A
B ∧ B = B

Answer 11

idempotent law

Question 12

(A’)’ = A
((A ∧ B)’)’ = A ∧ B (same for ∨)
(A ∨ B’)’ ≠ A ∨ B

Answer 12

Double negation law

Question 13

(A ∧ B)’ = A’ ∨ B’
(A ∨ B)’ = A’ ∧ B’

Answer 13

de Morgan’s Law

Question 14

A ∧ (A ∨ B) = A
A ∨ (A ∧ B) = A

Answer 14

Absorptive law

Question 15

Boolean Algebra is a way of formally specifying, or describing, a particular situation or procedure. We use variables to represent elements of our situation or procedure. Variables may take one of only two values. Traditionally this would be True and False. So for instance we may have a variable X and state that this represents if it is raining outside or not. The value of X would be :

Answer 15

True if it is raining outside.
False if it is not raining outside.

Question 16

When _______ are used in an expression this means that we evaluate that part of the expression first before the other parts.

Answer 16

brackets ( )

Question 17

If g is True and p is False then :

Substituting g and p for those values we get :

(True OR False) AND NOT(True AND False)
The first set of brackets (True OR False) AND NOT(True AND False) evaluates to True so let’s replace that into the expression and we get :

____ ____ ___ (___ __ ____)

The next set of brackets True AND NOT(True AND False) evaluates to _____ so let’s replace that into the expression as well giving us :

True AND NOT(False)

NOT(False) evaluates to True so we can apply that to the expression and we end up with :

True and True

And the final result is True.

Answer 17

True AND NOT(True AND False)
False

Question 18

NAND is effectively the opposite of what AND is.

r NAND S is equivalent to ___(__ ___ ___)

Answer 18

NOT(r AND s)

Question 19

NOR or NOT OR
NOR is effectively the opposite of OR.

b NOR k is equivalent to ___(__ ___ __)

Answer 19

NOT(b OR k)

Question 20

A term AND‘ed with a “0” equals 0 or OR‘ed with a “1” will equal 1

A . 0 = 0 A variable AND’ed with 0 is always equal to 0
A + 1 = 1 A variable OR’ed with 1 is always equal to 1

Answer 20

Annulment Law

Question 21

A term OR‘ed with a “0” or AND‘ed with a “1” will always equal that term

A + 0 = A A variable OR’ed with 0 is always equal to the variable
A . 1 = A A variable AND’ed with 1 is always equal to the variable

Answer 21

Identity Law

Question 22

An input that is AND‘ed or OR´ed with itself is equal to that input

A + A = A A variable OR’ed with itself is always equal to the variable
A . A = A A variable AND’ed with itself is always equal to the variable

Answer 22

Idempotent Law

Question 23

A term AND‘ed with its complement equals “0” and a term OR´ed with its complement equals “1”

A . A = 0 A variable AND’ed with its complement is always equal to 0
A + A = 1 A variable OR’ed with its complement is always equal to 1

Answer 23

Complement Law

Question 24

The order of application of two separate terms is not important

A . B = B . A The order in which two variables are AND’ed makes no difference
A + B = B + A The order in which two variables are OR’ed makes no difference

Answer 24

Commutative Law

Question 25

A term that is inverted twice is equal to the original term

A = A A double complement of a variable is always equal to the variable

Answer 25

Double Negation Law

Question 26

There are two “de Morgan’s” rules or theorems,

(1) Two separate terms NOR‘ed together is the same as the two terms inverted (Complement) and AND‘ed for example: A+B = _____

(2) Two separate terms NAND‘ed together is the same as the two terms ______ (Complement) and OR‘ed for example: A.B = A + B

Answer 26

AB
inverted

Question 27

While not Boolean Laws in their own right, these are a set of Mathematical Laws which can be used in the simplification of Boolean Expressions.

0 . 0 = 0 A 0 AND’ed with itself is always equal to 0
1 . 1 = 1 A 1 AND’ed with itself is always equal to 1
1 . 0 = 0 A 1 AND’ed with a 0 is equal to 0
0 + 0 = 0 A 0 OR’ed with itself is always equal to 0
1 + 1 = 1 A 1 OR’ed with itself is always equal to 1
1 + 0 = 1 A 1 OR’ed with a 0 is equal to 1
1 = 0 The Inverse (Complement) of a 1 is always equal to 0
0 = 1 The Inverse (Complement) of a 0 is always equal to 1

Answer 27

Boolean Postulates

Question 28

This law permits the multiplying or factoring out of an expression.

A(B + C) = A.B + A.C (OR _____ _____)
A + (B.C) = (A + B).(A + C) (AND ____ _____)

Answer 28

Distributive Law

Question 29

This law enables a reduction in a complicated expression to a simpler one by absorbing like terms.

A + (A.B) = (A.1) + (A.B) = A(1 + B) = A (OR Absorption Law)
A(A + B) = (A + 0).(A + B) = A + (0.B) = A (AND Absorption Law)

Answer 29

Absorptive Law

Question 30

This law allows the removal of brackets from an expression and regrouping of the variables.

A + (B + C) = (A + B) + C = A + B + C (OR Associate Law)
A(B.C) = (A.B)C = A . B . C (AND Associate Law)

Answer 30

Associative Law

Question 31

In mathematics, an _____ is a statement true for all possible values of its variable or variables.

The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original “anything,” no matter what value that “_______” (x) may be.

Like ordinary algebra, Boolean algebra has its own unique identities based on the bivalent states of Boolean variables.

Answer 31

identity
anything