Relations and Functions Test

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Relations and Functions Test - 18

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Weekly Quiz Competition

Question 1
1 / -0

A binary operation ∗ on a set A

A
is a function ∗ : A × A → A

B
adds 1 to all elements of A

C
function ∗: a + 4, a ∈ R .

D
multiplies all elements by a scalar

Solution

Let A be a non empty set .A function ∗ : A × A → A is called a binary operation on set A.
Question 2
1 / -0

A binary operation ∗ on the set X is called commutative, if

A
* is Transitive for every a, b ∈ X

B
a + b = b – a for every a, b ∈ X

C
∗ is open for every a, b ∈ X

D
a ∗ b = b ∗ a for every a, b ∈ X

Solution

A binary operation * on a set X is said to be commutative binary operation , if a * b = b * a for all a, b ∈ X
Question 3
1 / -0

If A is a non empty set, an element e ∈ A is called the identity for the binary operation ∗, if

A
a ∗ e = e = e ∗ a, ∀ a ∈ A

B
a ∗ e = a² = e ∗ a, ∃ a ∈ A

C
a ∗ e = a ≠ e ∗ a, ∀ a ∈ A

D
a ∗ e = a = e ∗ a, ∀ a ∈ A

Solution

Let * be a binary operation on a set A. If there exist an element e ∈ A such that a ∗ e = a = e ∗ a, ∀ a ∈ A. Then e is called an identity element for the binary operation * on A.
Question 4
1 / -0

An element a ∈ A is said to be invertible with respect to the operation ∗, if there exists

A
An element b in A such that a ∗ b = e = b ∗ a

B
An element b in A such that a ∗ b ∗ b = e ∗ a = b ∗ a ∗ a

C
An element b in A such that a ∗ b∗ a = e = b ∗ a

D
An element b in A such that a ∗ b∗ b = e = b ∗ a

Solution

Let * be a binary operation on a set A , and let e be an identity element for the binary operation * on A.Then an element a ∈ A is called an invertible element , If there exist an element b ∈ A such that a ∗ b = e = b ∗ a, then, b is called an inverse of element a.
Question 5
1 / -0

Division operation ∗ on the set R of nonzero real numbers is

A
a binary operation

B
commutative

C
a unary operation

D
not a binary operation

Solution

Division is well defined only for all non zero real numbers, therefore, Division operation ∗ on the set R of non zero numbers is a binary operation.
Question 6
1 / -0

∗: R × R → R given by (a , b) → a + 4b² ,is a

A
binary operation

B
unary operation

C
ternary operation

D
not well defined

Solution

∗: R × R → R given by (a , b) → a + 4b² is a binary operation, because (a , b) → a + 4b² is always real for all real values of a and b .
Question 7
1 / -0

+ : R × R → R and ×:R × R → R are

A
non associative operations

B
not commutative operations

C
non transitive operations

D
commutative operations

Solution

Since Addition and Multiplication are both commutative over the set of real numbers.
Question 8
1 / -0

A binary operation *: R × R → R defined by a ∗ b = a + 2b is

A
a unary operation

B
not commutative

C
Commutative

D
not well defined

Solution

Here, a * b = a + 2b , and b * a = b + 2a . Since ,a * b ≠ b * a. Therefore , * : R × R → R is not commutative.
Question 9
1 / -0

A binary operation * : R × R → R defined by a ∗ b = a + 2b is

A
A unary operation

B
Not well defined

C
Commutative

D
Not associative

Solution

Here, a * (b * c ) = a * ( b+ 2c) = a+ 2( b+2c) = a + 2b + 4c and (a *b)*c = (a + 2b) *c = a + 2b + 2c . Since a * (b * c) ≠ (a * b)* c. Therefore , * : R × R → R is not associative.
Question 10
1 / -0

If the binary operation ∗ defined on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of a and b, then ∗ is

A
not a binary operation

B
a binary operation

C
a unary operation

D
not well defined

Solution

We have , a ∗ b = L.C.M. of a and b ,where a,b ∈ {1,2,3,4,5} ∴ 1 * 1 = 1, 1 * 2 = 2, 1 * 3 = 3, 1 * 4 = 4, 1 * 5 = 5, 2 * 1 = 2, 2 * 2 = 2, 2 * 3 = 6, 2 * 4 = 4, 2 * 5 = 10 We observe that , 2 * 3 = 6 and 2 * 5 = 10 do not belong to the set {1,2,3,4,5}. So , * is not a binary operation on the given set.

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Relations and Functions Test - 18

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Relations and Functions Test - 18

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Question 1

is a function ∗ : A × A → A

adds 1 to all elements of A

function ∗: a + 4, a ∈ R .

multiplies all elements by a scalar

Solution

Question 2

* is Transitive for every a, b ∈ X

a + b = b – a for every a, b ∈ X

∗ is open for every a, b ∈ X

a ∗ b = b ∗ a for every a, b ∈ X

Solution

Question 3

a ∗ e = e = e ∗ a, ∀ a ∈ A

a ∗ e = a2 = e ∗ a, ∃ a ∈ A

a ∗ e = a ≠ e ∗ a, ∀ a ∈ A

a ∗ e = a = e ∗ a, ∀ a ∈ A

Solution

Question 4

An element b in A such that a ∗ b = e = b ∗ a

An element b in A such that a ∗ b ∗ b = e ∗ a = b ∗ a ∗ a

An element b in A such that a ∗ b∗ a = e = b ∗ a

An element b in A such that a ∗ b∗ b = e = b ∗ a

Solution

Question 5

a binary operation

commutative

a unary operation

not a binary operation

Solution

Question 6

binary operation

unary operation

ternary operation

not well defined

Solution

Question 7

non associative operations

not commutative operations

non transitive operations

commutative operations

Solution

Question 8

a unary operation

not commutative

Commutative

not well defined

Solution

Question 9

A unary operation

Not well defined

Commutative

Not associative

Solution

Question 10

not a binary operation

a binary operation

a unary operation

not well defined

Solution

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a ∗ e = a² = e ∗ a, ∃ a ∈ A