Properties of binary operations - class-XI

Question 1 Multiple Choice (Single Answer)

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows:
(i) $a\ast b = a - b$ (ii) $a\ast b = a^{2} + b^{2}$
(iii) $a\ast b = a + ab$ (iv) $a\ast b = (a - b)^{2}$
(v) $a\ast b = \dfrac {ab}{4}$ (vi) $a\ast b = ab^{2}$
Find which of the binary operations are commutative and which are associative

$ii, iv, v$ are commutative and $v$ associative
$ii, iv, v$ are not commutative and $v$ associative
$iii, iv, v$are commutative and $v$ associative
$vi, iv, v$are commutative and $v$ associative

$(i)$ $a\ast b=a-b$

Check commutative is

$a\ast b=b\ast a$

$a\ast b=a-b$

$b\ast a=b-a$

Since, $a\ast b\neq b\ast a$

$\ast$ is not commutative.

Check associative

$\ast$ is associative if

$(a\ast b)\ast c=a\ast (b\ast c)\ (a\ast b)\ast c={ (a-b) }^{ \ast }c=(a-b)-c=a-b-c\ a\ast (b\ast c)=a\ast (b-c)=a-(b-c)=a-b+c$

Since $ (a\ast b)\ast c\neq a\ast (b\ast c)$

$\ast$ is not an associative binary operation.

$(ii)$ $a\ast b={ a }^{ 2 }+{ b }^{ 2 }$

Check commutative

$\ast$ is commutative if $a\ast b=b\ast a$

$a\ast b={ a }^{ 2 }+{ b }^{ 2 }\ b\ast a={ b }^{ 2 }+{ a }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }$

Since $ a\ast b=b\ast a\quad \forall\quad a,b\epsilon Q$

$\ast$ is commutative.

Check associative

$\ast$ is associative if

$(a\ast b)\ast c=a\ast (b\ast c)\ (a\ast b)\ast c=({ a }^{ 2 }+{ b }^{ 2 })\ast c={ ({ a }^{ 2 }+{ b }^{ 2 }) }^{ 2 }+{ c }^{ 2 }\ a\ast (b\ast c)=a\ast ({ b }^{ 2 }+{ c }^{ 2 })={ a }^{ 2 }+{ ({ b }^{ 2 }+{ c }^{ 2 }) }^{ 2 }$

Since $ (a\ast b)\ast c\neq a\ast (b\ast c)$

$\ast$ is not an associative binary operation.

$(iii)$ $a\ast b=a+b$

Check commutative

$\ast$ is commutative is $a\ast b=b\ast a$

$ a\ast b=a+ab;\quad b\ast a=b+ba$

Since $ a\ast b\neq b\ast a$

$\ast$ is not commutative.

$(iv)$ $a\ast b={ (a-b) }^{ 2 }$

Check commutative

$\ast$ is commutative if $a\ast b=b\ast a$

$ a\ast b={ (a-b) }^{ 2 }\quad ;\quad b\ast a={ (b-a) }^{ 2 }={ (a-b) }^{ 2 }$

Since $ a\ast b=b\ast a\quad \forall\quad a,b\epsilon Q$

$\ast$ is commutative.

Check associative

$\ast$ if

$(a\ast b)\ast c=a\ast (b\ast c)\ (a\ast b)\ast c={ (a-b) }^{ 2 }\ast c={ [{ (a-b) }^{ 2 }-c] }^{ 2 }\ a\ast (b\ast c)=a\ast { (b-c) }^{ 2 }={ [a-{ (b-c) }^{ 2 }] }^{ 2 }$

Since $ (a\ast b)\ast c\neq a\ast (b\ast c)$

$\ast$ is not an associative binary operation.

$(v)$ $a\ast b=\cfrac { ab }{ 4 } $

Check commutative.

$\ast$ is commutative if $a\ast b=b\ast a$

$ a\ast b=\cfrac { ab }{ 4 } \quad ;\quad b\ast a=\cfrac { ba }{ 4 } =\cfrac { ab }{ 4 } $

Since $ a\ast b=b\ast a\quad \forall\quad a,b\epsilon Q$

$\ast$ is commutative.

Check associative.

$\ast$ is association if $(a\ast b)\ast c=a\ast (b\ast c)$

$(a\ast b)\ast c=(\cfrac { \cfrac { ab }{ 4 } \ast c }{ 4 } )=\cfrac { abc }{ 16 } \ a\ast (b\ast c)=a\ast (\cfrac { bc }{ 4 } )=\cfrac { a\times \cfrac { bc }{ 4 } }{ 4 } =\cfrac { abc }{ 16 } $

Since $ (a\ast b)\ast c=a\ast (b\ast c)\quad \forall\quad a,b,c\epsilon Q$

$\ast$ is an associative binary operation.

$(vi)$ $a\ast b={ ab }^{ 2 }$

check commutative.

$\ast$ is commutative if $a\ast b=b\ast a$

$ a\ast b={ ab }^{ 2 }\quad ;\quad b\ast a={ ba }^{ 2 }$

Since $ a\ast b\neq b\ast a$

$\ast$ is not commutative.

Check associative

$\ast$ is associative if $(a\ast b)\ast c=a\ast (b\ast c)$

$(a\ast b)\ast c={ ab }^{ 2 }\ast c=({ ab }^{ 2 }){ c }^{ 2 }=a{ b }^{ 2 }{ c }^{ 2 }.\ a\ast (b\ast c)=a\ast { bc }^{ 2 }=a{ ({ bc }^{ 2 }) }^{ 2 }=a{ b }^{ 2 }{ c }^{ 4 }$

Since $ (a\ast b)\ast c\neq a\ast (b\ast c)$

$\ast$ is not an associate binary operation.

Question 2 Multiple Choice (Single Answer)

State whether the following statements are true of false. Justify.
(i) For an arbitrary binary operation $\ast$ on as set $N, a\ast a = a\forall a \epsilon N$
(ii) If $\ast$ is a commutative binary operation on $N$, then $a\ast (b\ast c) = (c\ast b) \ast a$

True
False

Question 3 Multiple Choice (Single Answer)

Consider a binary operation $\ast$ on $N$ defined as $a\ast b = a^{3} + b^{3}$. Choose the correct answer

Is $\ast$ both associative and commutative?
Is $\ast$ commutative but not associative?
s $\ast$ associative but not commutative?
Is $\ast$ neither commutative nor associative?

Question 4 Multiple Choice (Single Answer)

Let $$ be a binary operation defined on the set of rational numbers $Q$ defined by $a * b= ab + 1,$ in this statement $$ is a commutative.

True
False

Question 5 Multiple Choice (Single Answer)

Sum of $(267 + 345) + 21$ and $267 + (345 + 21)$ will be same.

True
False

Question 6 Multiple Choice (Single Answer)

The set of integers $Z$ with the binary operation $*$ defined as $a * b = a + b+ 1$ for $a, b, Z$ is a group. The identity element of this group is

$0$
$1$
$-1$
$15$

Question 7 Multiple Choice (Single Answer)

If the binary operation $*$ is defined on a set of ordered pairs of real numbers as $(a, b) * (c, d) = (a \times d + b \times c, b \times d)$ and is associative, then $(1, 2) * (3, 5) * (3, 4)$ is equal to

$(74,40)$
$(32,40)$
$(23,11)$
$(7,11)$

Question 8 Multiple Choice (Single Answer)

The set of all real numbers under the usual multiplication operation is not a group since

multiplication is not a binary operation
multiplication is not associative
identity element does not exist
zero has no inverse

Question 9 Multiple Choice (Single Answer)

If * is defined on the set R of all real numbers by $a*b=\sqrt{a^2+b^2}$, find the identity element in R with respect to *.

0
1
2
3

Question 10 Multiple Choice (Single Answer)

Subtraction of integers is an operation that is

commutative and associative
not commutative but associative
neither commutative nor associative
commutative but not associative.

Question 11 Multiple Choice (Single Answer)

A closed set with respect to some binary operation is called semi- group if

$*$ is associative
$*$ is commutative
$*$ is anti-commutative
identity element exists

Question 12 Multiple Choice (Single Answer)

$"*"$ is said to be commutative in $A$ for all $a,b \epsilon A$

$a+b=b+a$
$a*b=b*a$
$a-b=b-a$
$a*b\neq b*a$

Question 13 Multiple Choice (Single Answer)

If $A \ast B = A \cap B$ on $P(X)$, then identify for $\ast$ is ________$(X \neq \phi)$

$\phi$
$X$
$U$
$A$

Properties of binary operations - class-XI

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