Binary Operations Concepts Flashcards
a. If ∗ is any binary operation on any set S, then a ∗ a = a for all a ∈ S.
False
b. If ∗ is any commutative binary operation on any set S, then a ∗ (b ∗ c) = (b ∗ c) ∗ a for all a, b, c ∈ S
True
c. If ∗ is any associative binary operation on any set S, then a ∗ (b ∗ c) = (b ∗ c) ∗ a for all a, b, c ∈ S.
F
d. The only binary operations of any importance are those defined on sets of numbers.
False
e. A binary operation ∗ on a set S is commutative if there exists a, b ∈ S such that a ∗ b = b ∗ a.
Fasle
f. Every binary operation defined on a set having exactly one element is both commutative and associative.
T
g. A binary operation on a set S assigns at least one element of S to each ordered pair of elements of S.
True
h. A binary operation on a set S assigns at most one element of S to each ordered pair of elements of S
T
i. A binary operation on a set S assigns exactly one element of S to each ordered pair of elements of S.
T
j. A binary operation on a set S may assign more than one element of S to some ordered pair of elements of S.
F
k. For any binary operation ∗ on the set S, if a, b, c ∈ S, and a ∗ b = a ∗ c, then b = c.
F
l. For any binary operation ∗ on the set S, there is an element e ∈ S such that for all x ∈ S, x ∗ e = x.
F
m. There is an operation on the set S={e1,e2, a} so that for all x ∈ S, e1 ∗ x = e2∗ x = x.
T
m. There is an operation on the set S={e1,e2, a} so that for all x ∈ S, e1 ∗ x = e2∗ x = x.
T
n. Identity elements are always called e.
F
A BO * on a set S is commutative iff ab=ba for all a,b∈S
A BO * on a set S is commutative iff ab=ba for all a,b∈S
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).
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).
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.
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.
An identity is the set S with operation * is element e∈S such that for all a∈S, ae=ea=a
An identity is the set S with operation * is element e∈S such that for all a∈S, ae=ea=a
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.
No, because e1e2=e1 and e1e2=e2
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.
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
Every binary operation on a set consisting of a single element is both commutative and associative.
True, BO with one element, that element is the result of any computation
Function subtraction − on F is commutative.
Not Commutative
find an example
Function subtraction − on F is associative.
It is not associative. find an example
Under function subtraction − F has an identity.
no identity
Under function multiplication · F has an identity.
The constant f(x)=1 is an identity element in F
Function multiplication · on F is commutative.
True, find an example
Function multiplication · on F is associative.
True, find an example
Function composition ◦ on F is commutative.
f◦g/=g◦f
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.
it is not true. Let *be + and *’ be . let S=Z. Then 2+(3.5)=17 but (2+3).(2+5)=35
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 ∗.
Let a,b ∈ H. by the definition of H, we have ax=xa and bx=xb for all x∈S. Using the fact that * is associative we then obtain for all x∈S,
(ab)x=a(bx)=a(xb)=(ax)b=(xa)b=x(ab)
This show ab satifies the defining criterion for an element of H, so (ab)∈H
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.
For any x, y∈S,xy=(xy)(xy)=(xy)x)y=(yx)x)y=((xx) y) * y=(xy)y=(yy)x= yx. So, is commutative. Since * is commutative, (x y)z = (y * 2) * x = x * (y * 2) for and x, y∈S. So, * isassociative.
