Lec2. Boolean algebra

Question 1

A function from S to S is also called a ____ on S

Answer 1

unary operation

Question 2

A function from S^2 to S is called a _____ on S

Answer 2

binary operation

Question 3

True/False: unary operation and binary operation assuming that DOD = domain

Answer 3

true

Question 4

notion for unary operation?

Answer 4

~

Question 5

True/False: ~u= u^ ~ = ~(u)

Answer 5

false - the first 2 is true but the last one is false

Question 6

Notation for binary operation?

Answer 6

any symbol

Question 7

True/False: u * v = *((u,v))

Answer 7

False

Question 8

State the idempotent property

Answer 8

for any u in S we have u★u=u

Question 9

State the commutative property

Answer 9

for any (u,v0 in S^2 we have u★v=v★u

Question 11

State the associate property

Answer 11

for any (u,v,w) in S^3 we have u★(v★w) =(u★v)★w

Question 12

Example of commutative and associative

Answer 12

+, x, min, and max

Question 13

Example of idempotent

Answer 13

average of 2 equal numbers, min, max

Question 14

State the distribute property: ★ is distribute over *

Answer 14

for any (u,v,w) in S^3 we have u★ (vw) = (u★v) * (u ★ w) and (vw) ★ u = (vu) ★ (wu)

Question 15

Example of distributive

Answer 15

x is distribute over +, min is distribute over max and vice versa

Question 16

State the neutral element

Answer 16

n is a neutral / zero element for ★ if n ★ u = u ★ n = u for any u in S

Question 17

State the absorbing element

Answer 17

a is a absorbing / unit element for ★ if u ★ a = u ★ a = a for any u in S

Question 18

True/False: If m∈S and n∈S are neutral elements for ★ then m=n.

Answer 18

true

Question 19

True/False: If a∈S and b∈S are absorbing elements for ★ then a=b.

Answer 19

true

Question 20

precedence rule?

Answer 20

bracket > unary > binary

Question 21

Defining a boolean algebra (B, + , ^ , -)

Answer 21

4 tuple
B is a set (of anything)
+ and ^ are binary operations on B
- is a unary operation on B
there are 2 distinct elements 0 and 1 such that 0 is a neutral element for + and 1 is a neutral element for ^

Question 22

fundamental law of boolean algebra

Answer 22

identity, commutative, associative, distributive, complement

Question 23

Derived from fundamental law

Answer 23

double complement, domination, idempotent, de Morgans

Question 24

What is boolean value?

Answer 24

an element of B

Question 25

What is a boolean variable?

Answer 25

variable that represents an element of B

Question 26

Domain of the boolean function is?

Answer 26

2^k = # rows in a table

Question 27

Range of a boolean function?

Answer 27

{0,1}

Question 28

define literal?

Answer 28

boolean variable or its complement

Question 29

define minterm of degree n ∈ N*?

Answer 29

product of n literals
all input must appear but ONCE time only

Question 30

define maxterm of degree n ∈ N*?

Answer 30

sum of n literals

Question 31

define CNF, and what violate CNF form?

Answer 31

product of sum literal
(d1) (d2) (d3)
- complement can only apply to single variable
- no multiplication within a clause

Question 32

define DNF, and what violate DNF form?

Answer 32

sum of product literal
c1 + c2 + c3
- complement can only apply to single variable
- no addition within a term

Question 33

What is boolean function?

Answer 33

adding all the midterms (like DNF)

Question 34

What is functionally complete?

Answer 34

{addition, complement}, {multiplication, complement}, NAND, NOR

Question 35

True/False: u and u^ - are minterm of degree 1

Answer 35

true

Question 36

True/False: u + v ^- are minterm of degree 2

Answer 36

false, maxterm

Question 37

True/False: u. v. w are minterm of degree 1

Answer 37

false, degree 3

Question 38

True/False: u + u^- are maxterm of degree 2

Answer 38

false, neither minterm nor maxterm since 2 literals for the same variable

Question 39

True/False: (u.v)^- are minterm of degree 2

Answer 39

false, not minterm nor maxterm since that is not a literal

Question 40

When 2 boolean expressions are equivalent?

Answer 40

yielding the same truth table

Question 41

duality principle

Answer 41

interchanging unit with zero element and a binary expression with another one

Question 42

duality and equivalent

Answer 42

if the 2 expressions are equivalent, then they are dual expressions also equivalent