Binary Operations Concepts

Question 1

a. If ∗ is any binary operation on any set S, then a ∗ a = a for all a ∈ S.

Answer 1

False

Question 2

b. If ∗ is any commutative binary operation on any set S, then a ∗ (b ∗ c) = (b ∗ c) ∗ a for all a, b, c ∈ S

Answer 2

True

Question 3

c. If ∗ is any associative binary operation on any set S, then a ∗ (b ∗ c) = (b ∗ c) ∗ a for all a, b, c ∈ S.

Answer 3

F

Question 4

d. The only binary operations of any importance are those defined on sets of numbers.

Answer 4

False

Question 5

e. A binary operation ∗ on a set S is commutative if there exists a, b ∈ S such that a ∗ b = b ∗ a.

Answer 5

Fasle

Question 6

f. Every binary operation defined on a set having exactly one element is both commutative and associative.

Answer 6

T

Question 7

g. A binary operation on a set S assigns at least one element of S to each ordered pair of elements of S.

Answer 7

True

Question 8

h. A binary operation on a set S assigns at most one element of S to each ordered pair of elements of S

Answer 8

T

Question 9

i. A binary operation on a set S assigns exactly one element of S to each ordered pair of elements of S.

Answer 9

T

Question 10

j. A binary operation on a set S may assign more than one element of S to some ordered pair of elements of S.

Answer 10

F

Question 11

k. For any binary operation ∗ on the set S, if a, b, c ∈ S, and a ∗ b = a ∗ c, then b = c.

Answer 11

F

Question 12

l. For any binary operation ∗ on the set S, there is an element e ∈ S such that for all x ∈ S, x ∗ e = x.

Answer 12

F

Question 13

m. There is an operation on the set S={e1,e2, a} so that for all x ∈ S, e1 ∗ x = e2∗ x = x.

Answer 13

T

Question 14

m. There is an operation on the set S={e1,e2, a} so that for all x ∈ S, e1 ∗ x = e2∗ x = x.

Answer 14

T

Question 15

n. Identity elements are always called e.

Answer 15

F

Question 16

􏰫 􏰌􏰎􏰉􏰄􏰈􏰙 􏰇􏰋􏰃􏰈􏰄􏰆􏰎􏰇􏰉 􏲈 􏰇􏰉 􏰄􏰫 􏰌􏰎􏰉􏰄􏰈􏰙 􏰇􏰋􏰃􏰈􏰄􏰆􏰎􏰇A BO * on a set S is commutative iff ab=ba for all a,b∈S

Answer 16

􏰇􏰉 􏰄􏰫 􏰌􏰎􏰉􏰄􏰈􏰙 􏰇􏰋􏰃􏰈􏰄􏰆􏰎􏰇A BO * on a set S is commutative iff ab=ba for all a,b∈S

Question 17

A binary operation ∗ on a set S is associative if and only if, for all a, b, c ∈ S, we have
(b∗c)∗a = b∗(c∗a).

Answer 17

A binary operation ∗ on a set S is associative if and only if, for all a, b, c ∈ S, we have
(b∗c)∗a = b∗(c∗a).

Question 18

A subset H of a set S is closed under a binary operation∗ on S if and only if (a∗b)∈H for all a,b ∈H.

Answer 18

A subset H of a set S isclosed under a binary operation∗ on S if and only if (a∗b)∈H for all a,b ∈H.

Question 19

􏰫􏰉 􏰎􏰊􏰃􏰉􏰆􏰎􏰆􏰙 􏰎􏰉 􏰆􏰂􏰃 􏰏􏰃An identity is the set S with operation * is element e∈S such that for all a∈S, ae=ea=a

Answer 19

An identity is the set S with operation * is element e∈S such that for all a∈S, ae=ea=a

Question 20

Is there an example of a set S, a binary operation on S, and two different elements e1, e2 ∈ S such that for all a∈S,e1 ∗a=a and a∗e2 =a? If so, give an example and if not, prove there is not one.

Answer 20

No, because e1e2=e1 and e1e2=e2

Question 21

Prove that if ∗ is an associative and commutative binary operation on a set S, then
(a ∗ b) ∗ (c ∗ d) = [(d ∗ c) ∗ a] ∗ b
for all a, b, c, d ∈ S.
Assume the associative law only for triples as in the definition, that is, assume only

(x ∗ y) ∗ z = x ∗ (y ∗ z)
for all x,y,z ∈ S.

Answer 21

We have
(ab)(cd)=(cd)(ab)=(dc)(ab)
=[(dc)a]b
Where we used commutativity for the first two steps and associativity for the last

Question 22

Every binary operation on a set consisting of a single element is both commutative and associative.

Answer 22

True, BO with one element, that element is the result of any computation

Question 23

Function addition + on F is associative.

Answer 23

It is,
(f+g)+h=f+(g+h)

Question 24

Function subtraction − on F is commutative.

Answer 24

Not Commutative
find an example

Question 25

Function subtraction − on F is associative.

Answer 25

It is not associative. find an example

Question 26

Under function subtraction − F has an identity.

Answer 26

no identity

Question 27

Under function multiplication · F has an identity.

Answer 27

The constant f(x)=1 is an identity element in F

Question 28

Function multiplication · on F is commutative.

Answer 28

True, find an example

Question 29

Function multiplication · on F is associative.

Answer 29

True, find an example

Question 30

Function composition ◦ on F is commutative.

Answer 30

f◦g/=g◦f

Question 31

If ∗ and ∗′ are any two binary operations on a set S, then a∗(b∗′ c)=(a∗b)∗′ (a∗c), for all a,b,c∈S.

Answer 31

it is not true. Let *be + and *' be . let S=Z. Then 2+(3.5)=17 but (2+3).(2+5)=35

Question 32

Suppose that∗is an associative binary operation on a set S. Let H {a∈S|a∗x=x∗aforallx∈S}. Show that H is closed under ∗.

Answer 32

Let a,b ∈ H. by the definition of H, we have a*x=x*a and b*x=x*b for all x∈S. Using the fact that * is associative we then obtain for all x∈S, (a*b)*x=a*(b*x)=a*(x*b)=(a*x)*b=(x*a)*b=x*(a*b) This show a*b satifies the defining criterion for an element of H, so (a*b)∈H

Question 33

Let S be a set and let ∗ be a binary operation on S satisfying the two laws x∗x =xfor all x∈S,and (x∗y)∗z=(y∗z)∗x for all x,y,z ∈ S.

Answer 33

For any x, y∈S,x*y=(x*y)*(x*y)=(x*y)*x)*y=(y*x)*x)*y=((x*x) *y) * y=(x*y)*y=(y*y)*x= y*x. So, *is commutative. Since * is commutative, (x* y)*z = (y * 2) * x = x * (y * 2) for and x, y∈S. So, * isassociative.