##### Question 1:

Determine whether each of the following relations are reflexive, symmetric and transitive : Relation R in the set A = {1, 2, 3, ......, 13, 14} defined as : R = {(x, y) : 3x – y = 0}

##### Answer:

Here

A = {1, 2, 3, ....... , 13, 14}

and R = {(x, y) : 3x – y = 0}

##### Question 2:

Determine whether each of the following relations
are reflexive, symmetric and transitive :

Relation R in the set N of natural numbers defined
as :

R = {(x, y) : y = x + 5 and x < 4}

##### Answer:

##### Question 3:

Determine whether each of the following relations
are reflexive, symmetric and transitive :

Relation R in the set A = {1, 2, 3, 4, 5, 6} defined
as :

R = {(x, y) : y is divisible by x }

##### Answer:

##### Question 4:

Determine whether each of the following relations
are reflexive, symmetric and transitive :

Relation R in the set Z of all integers defined as :

R = {(x, y) : x – y is an integer}

##### Answer:

##### Question 5:

Determine whether each of the following relations
are reflexive, symmetric and transitive :

Relation R in the set A of human beings in a town
at a particular time given by :

(a) R = {(x, y) : x and y work at the same place}

(b) R = {(x, y) : x and y live in the same locality}

(c) R = {(x, y) : x is exactly 7 cm taller than y}

(d) R = {(x, y) : x is wife of y}

(e) R = {(x, y) : x is father of y}.

##### Answer:

##### Question 6:

Show that the relation R in the set R of real
numbers, defined as :

R = {(a, b) : a $\le $ b^{2}}
is neither reflexive nor symmetric nor transitive.

##### Answer:

R is not reflexive.

##### Question 7:

Check whether the relation R defined in the set
{1, 2, 3, 4, 5, 6} as :

R = {(a, b) : b = a + 1}, is reflexive, symmetric or
transitive.

##### Answer:

##### Question 8:

Show that the relation R in R defined as :

R = {(a, b) : a $\le $ b}

is reflexive and transitive but not symmetric.

##### Answer:

##### Question 9:

Check whether the relation R in R defined by :

R = {(a, b) : a $\le $ b^{3}}

is reflexive, symmetric or transitive.

##### Answer:

R is not reflexive.

R is not symmetric.

[$\therefore $ If a $\le $ b^{3}, then b is not less than or equal to a^{3}

e.g. 1 $\le $ 3^{3}. But 3 is not less then 1^{3}]

R is not transitive.

[$\therefore $ If a $\le $ b^{3} and b $\le $ c^{3}, then a is not necessarily less
than or equal to c^{3}

e.g. Take a = 7, b = 2, c = 1·5.

Here a < b^{3} as 7 < 2^{3} = 8

b < c^{3} as 2 < (1·5)^{3} = 3.375

But a > c^{3} as 7 > (1·5)^{3} = 3·375]

##### Question 10:

Show that the relation R in the set {1, 2, 3} defined as :

R = {(1, 2), (2, 1)}, is symmetric but neither reflexive nor transitive.

##### Answer:

Here R = {(1, 2), (2, 1)}.

R is not reflexive. [$\therefore $ (1, 1), (2, 2), (3, 3) $\notin $ R]

R is not transitive.

[$\therefore $ (1, 2) $\in $ R, (2, 1) $\in $ R but (1, 1) $\notin $ R]

R is symmetric. [$\therefore $ (1, 2) $\in $ R and (2, 1) $\in $ R]

##### Question 11:

Show that the relation R in the set A of all books in a library of a college, given by :

R = {(x, y) : x and y have same number of pages} is an equivalence relation.

##### Answer:

We have :

##### Question 12:

Show that the relation in the set A = {1, 2, 3, 4, 5}, given by :

R = {(a, b) : | a – b | is even} is an equivalence relation.

Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

##### Answer:

##### Question 13:

Show that each of the relation R in the set :

A = {x : x $\in $ Z, 0 $\le $ x $\le $ 12}, given by :

(i) R = {(a, b) : | a – b | is a multiple of 4}

(ii) R = {(a, b) : a = b}

is an equivalence relation.

Find the set of all elements related to 1 in each case.

##### Answer:

Here

##### Question 14:

Give an example of a relation, which is :

Symmetric but neither reflexive nor transitive

##### Answer:

Let A = {1, 2, 3}.

The relation R = {(2, 3), (3, 2)} is symmetric but neither
reflexive nor transitive.

[$\therefore $ (1, 1) $\notin $ R; (2, 3), (3, 2) $\in $ R but (2, 2) $\notin $ R]

##### Question 15:

Give an example of a relation, which is :

Transitive but neither reflexive nor symmetric

##### Answer:

Let A = {1, 2, 3}.

The relation R = {(1, 3), (3, 2), (1, 2)} is transitive but
neither reflexive nor symmetric.

[$\therefore $ (1, 1) $\notin $ R; (1, 3) $\in $ R but (3, 1) $\notin $ R]

##### Question 16:

Give an example of a relation, which is :

Reflexive and symmetric but not transitive

##### Answer:

Let A = {1, 2, 3}.

The relation R = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2), (1,
2), (2, 1)}

is reflexive and symmetric but not transitive.

[$\therefore $ (1, 2) and (2, 3) $\in $ R but (1, 3) $\notin $ R]

##### Question 17:

Give an example of a relation, which is :

Reflexive and transitive but not symmetric

##### Answer:

Let A = {1, 2, 3}.

The relation R = {(1, 1), (2, 2), (3, 3), (1, 2)} is reflexive
and transitive but not symmetric.

[$\therefore $ (1, 2) $\in $ R but (2, 1) $\notin $ R]

##### Question 18:

Give an example of a relation, which is :

Symmetric and transitive but not reflexive

##### Answer:

Let A = {1, 2, 3}.

The relation R = {(1, 2), (2, 1), (1, 1), (2, 2)} is symmetric
and transitive but not reflexive. [$\therefore $ (3, 3) $\notin $ R]

##### Question 19:

Show that the relation R in the set A of points in a
plane, given by :

R = {(P, Q) : distance of the point P from the origin is
same as the distance of the point Q from the origin}
is an equivalence relation.

Further, show that the set of all points related to a point P $\ne $ (0, 0) is the circle passing through A with origin as centre.

##### Answer:

##### Question 20:

Show that the relation R, defined by the set A of
all triangles as :

R = {(T_{1}, T_{2}) : T_{1} is similar to T_{2}] is an equivalence
relation.

Consider three right angle triangles T_{1} with sides 3, 4,
5; T_{2} with sides 5, 12, 13 and T_{3} with sides 6, 8, 10.

Which triangles among T_{1}, T_{2} and T_{3} are related?

##### Answer:

Here

$\Rightarrow $ T_{1}is related to T

_{3}and T

_{3}is related to T

_{1}

$\Rightarrow $ (T

_{1}, T

_{3}) $\in $ R.

##### Question 21:

Show that the relation related to R, defined in the set of all polygons as :

R = {(P_{1}, P_{2}), P_{1} and P_{2} have same number of sides}
is an equivalence relation.

What is the set of all elements in A related to the right triangle T with sides 3, 4 and 5 ?

##### Answer:

##### Question 22:

Let L be the set of all lines in XY-plane and R is
the relation in L defined as :

R = {(L_{1}, L_{2}) : L_{1} is parallel to L_{2}}.

Show that R is an equivalence relation.

Find the set of all lines related to the line y=2x + 4.

##### Answer:

We have :

##### Question 23:

Choose the correct answer :

Let R be the relation in the set {1, 2, 3, 4} given

by : R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.

- R is reflexive and symmetric but not transitive
- R is reflexive and transitive but not symmetric
- R is symmetric and transitive but not reflexive
- R is an equivalence relation

##### Answer:

R is reflexive and transitive but not symmetric

##### Question 24:

Choose the correct answer :

Let R be the relation in the set N given by :

R = {(a, b) : a = b – 2, b > 6}.

- (2, 4) $\in $ R
- (3, 8) $\in $ R
- (6, 8) $\in $ R
- (8, 7) $\in $ R.

##### Answer:

(6, 8) $\in $ R

##### Question 25:

Show that the function f : R_{*} $\to $ R_{*} defined by

is one-one and onto, where R_{*} is the set of all nonzero
real numbers. Is the result true if the domain R_{*} is
replaced by N with co-domain having same as R_{*} ?

##### Answer:

Let x_{1}, x_{2} $\in $ R_{*}.

##### Question 26:

Check the injectivity and surjectivity of the following
function :

f : N $\to $ N given by f (x) = x^{2}

##### Answer:

Let x_{1}, x_{2} $\in $ N.

$\Rightarrow $ − + (x_{2} x_{1} ) (x_{2} x_{1} ) = 0 $\Rightarrow $ x_{2} – x_{1} = 0

[$\therefore $ x_{1}, x_{2} $\in $ N $\therefore $ x_{1} + x_{2} $\ne $ 0]

$\Rightarrow $ x_{1} = x_{2} $\Rightarrow $ f is one-one

$\Rightarrow $ f is injective.

Now range of f = {1^{2}, 2^{2}, 3^{2}, ...}

= {1, 4, 9, ...} $\ne $ N

$\Rightarrow $ f is not onto $\Rightarrow $ f is not surjective.

##### Question 27:

Check the injectivity and surjectivity of the following
function :

f : Z $\to $ Z given by f (x) = x^{2}

##### Answer:

Let x_{1}, x_{2} $\in $ Z.

$\Rightarrow $ x_{1} = x_{2} or x_{1} = – x_{2}

$\Rightarrow $ x_{2} = x_{1} or x_{2} = – x_{1}.

Thus f (x_{1}) = f (– x_{1}) −∨

x_{1} $\in $ Z

$\Rightarrow $ f is not one-one $\Rightarrow $ f is not injective.

Also range of f = {0, 1^{2}, 2^{2}, ...} = {0, 1, 4, ...} $\ne $ Z

$\Rightarrow $ f is not onto $\Rightarrow $ f is not surjective.

##### Question 28:

Check the injectivity and surjectivity of the following
function :

f : R $\to $ R given by f (x) = x^{2}

##### Answer:

Let x_{1}, x_{2} $\in $ R.

As in part (ii), f is not injective.

Also range of f = {0, 1^{2}, 2^{2}, ...}

= {0, 1, 4, ...} $\ne $ R

$\Rightarrow $ f is not onto $\Rightarrow $ f is not surjective.

##### Question 29:

Check the injectivity and surjectivity of the following
function :

f : N $\to $ N given by f (x) = x^{3}

##### Answer:

Let x_{1}, x_{2} $\in $ N.

$\Rightarrow $ f is one-one $\Rightarrow $ f is injective.

Now range of f = {1^{3}, 2^{3}, 3^{3}, ...} = {1, 8, 27, ...} $\ne $ N

$\Rightarrow $ f is not onto $\Rightarrow $ f is not surjective.

##### Question 30:

Check the injectivity and surjectivity of the following
function :

f : Z $\to $ Z given by f (x) = x^{3}

##### Answer:

Let x_{1}, x_{2} $\in $ Z.

As in part (iv), f is injective.

Now range of f = {1^{3}, 2^{3}, 3^{3}, ...} = {1, 8, 27, ...} $\ne $ Z

$\Rightarrow $ f is not onto $\Rightarrow $ f is not surjective.

##### Question 31:

Show that the Greatest Integer Function f : R $\to $ R
given by :

f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

##### Answer:

##### Question 32:

Show that the Modulus Function f : R $\to $ R, given
by :

f (x) = | x |

is neither one-one nor onto, where | x | is x, if x is positive and | x | is – x, if x is negative.

##### Answer:

##### Question 33:

Show that the Signum function f : R $\to $ R given by :

is neither one-one nor onto.##### Answer:

Since f (x) =1 for all x $\in $ (0, $\infty $)

and f (x) = – 1 for all x $\in $ (– $\infty $, 0),

$\therefore $ f is many-one $\Rightarrow $ f is not one-one.

Also range of f = {– 1, 0, 1) $\ne $ R

$\Rightarrow $ f is not onto.

##### Question 34:

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4),

(2, 5), (3, 6)} be a function from A to B. Show that f is oneone.

##### Answer:

We have : f = {(1, 4), (2, 5), (3, 6)}.

Then f (1) = 4, f (2) = 5 and f (3) = 6

$\Rightarrow $ Different points of domain correspond to different
f-images in the range.

Hence, f is one-one.

##### Question 35:

In each of the following cases, state whether the
function is one-one, onto or bijective. Justify your answer.

(i) f : R $\to $ R defined by f (x) = 3 – 4x

(ii) f : R $\to $ R defined by f (x) = 1 + x^{2}.

##### Answer:

(i) Let x_{1}, x_{2} $\in $ R.

Now f (x_{1}) = f (x_{2})

$\Rightarrow $ 3 – 4x1 = 3 – 4x2

$\Rightarrow $ x_{1} = x_{2} $\Rightarrow $ f is one-one.

Let y $\in $ R. Let y = f (x_{0}).

$\therefore $ For each y $\in $ R, there exists x0 $\in $ R such that
f (x0) = y.

$\therefore $ f is onto.

Hence, ‘f’ is one-one and onto or bijective.

(ii) Here f (1) = 1 + 1 = 2,

f (– 1) = 1 + 1 = 2.

Now 1 $\ne $ – 1 but f (1) = f (– 1)

$\therefore $ f is not one-one.

Also range of f is [1, $\infty $) $\ne $ R

$\therefore $ f is not onto.

Hence, ‘f’ is not bijective.

##### Question 36:

Let A and B be sets. Show that :

f : A × B $\to $ B × A such that f (a, b) = (b, a) is a bijective function.

##### Answer:

(a_{1}, b_{1}), (a_{2}, b_{2}) $\in $ A × B such that

f (a_{1} , b_{1} ) = (a_{2}, b_{2} )

$\Rightarrow $ (b_{1} , a_{1} ) = (b_{2}, a_{2} )

$\Rightarrow $ b_{1} = b_{2} and = a_{1} = a_{2}

$\Rightarrow $ (a_{2}, b_{2} ) = (a_{1} , b_{1} )

$\therefore $ f is one-one.

And corresponding to each ordered pair (y, x) $\in $ B × A,

there exists (x, y) $\in $ (A × B) such that

f (x, y) = (y, x)

$\therefore $ f is onto.

Hence, ‘f ’is a bijective function.

##### Question 37:

Let f : N $\to $ N be defined by :

State whether the function f is onto, one-one or bijective. Justify your answer.

##### Answer:

##### Question 38:

Let A = R – {3} and B = R – {1}.

Consider the function f : A $\to $ B defined by :

##### Answer:

Let x_{1}, x_{2} $\in $ R – {3}.

Now 1 f (x ) = f (x_{2} )

$\Rightarrow $ f is onto.

Hence, ‘f’ is one-one and onto.

##### Question 39:

Choose the correct answer :

Let f : R $\to $ R be defined as f (x) = x^{4}.

- f is one-one onto
- f is many-one onto
- f is one-one but not onto
- f is neither one-one nor onto.

##### Answer:

f is neither one-one nor onto.

##### Question 40:

Choose the correct answer.

Let f : R $\to $ R be defined as f (x) = 3x.

- f is one-one onto
- f is many one onto
- f is one-one but not onto
- f is neither one-one nor onto.

##### Answer:

f is one-one onto

##### Question 41:

Let f : {1, 3, 4} $\to $ {1, 2, 5} and g : {1, 2, 5} $\to $ {1, 3}

be given by :

f = {(1, 2), (3, 5), (4, 1)

and g = {(1, 3), (2, 3), (5, 1)}.

Write down gof.

##### Answer:

We have : f : {1, 3, 4} $\to $ {1, 2, 5}

and g : {1, 2, 5} $\to $ {1, 3}.

Here f (1) = 2, f (3) = 5 and f (4) = 1

and g (1) = 3, g (2) = 3 and g (5) = 1.

Here Rf = {1, 2, 5} = Dg

$\therefore $ Dgof = Df = {1, 3, 4}.

Now (gof) (1) = g (f (1)) = g (2) = 3

(gof) (3) = g (f (3)) = g (5) = 1

and (gof) (4) = g (f (4)) = g (1) = 3.

Hence, (gof) : {(1, 3), (3, 1), (4, 3)}.

##### Question 42:

Let f, g and h be functions from R to R. Show
that :

(i) (f + g) oh = foh + goh

(ii) (f.g) oh = (foh) . (goh).

##### Answer:

(i) For all x $\in $ R,

[( f + g) oh] (x) = ( f + g) (h(x))

= f (h(x))+g(h(x))= ( foh) (x)+(goh)(x)

=[( foh)+(goh)] (x).

Hence, ( f + g) oh = foh+ goh.

(ii) For all x $\in $ R,

(( f .g) oh) x = ( f .g) (h(x))

= f (h(x)) g (h(x))

= ( foh) (x) . (goh) (x)

= [( foh) . (goh)] (x).

Hence ( f .g) oh = ( foh) . (goh) .

##### Question 43:

Find gof and fog, if :

(i) f (x) = | x | and g (x) = | 5x – 2 |

(ii) f (x) = 8x^{3} and g (x) = x^{1/3}.

##### Answer:

(i) (a) gof (x)= g ( f (x))

= g (|x|) = |5|x|−2|.

(b) fog (x) = f (g (x)) = f (|5x −2|)

= |5x −2|.

(ii) (a) gof (x) = g ( f (x)) = g(8x^{3} )

=(8x^{3})^{1/3} =2x .

(b) fog (x) = f (g (x))

f(x^{1/3}) = 8(x^{1/3})^{3} =8x

##### Question 44:

##### Answer:

##### Question 45:

State with reasons whether following functions have
inverse :

(i) f : {1, 2, 3, 4} $\to $ {10} with

f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} $\to $ {1, 2, 3, 4} with

g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} $\to $ {7, 9, 11, 13} with

h = {(2, 7), (3, 9), (4, 11), (5, 13)}.

##### Answer:

(i) Here f (1) = f (2) = f (3) = f (4) = 10,

$\therefore $ f is not one-one

[$\therefore $1, 2, 3, 4 have same image 10]

$\Rightarrow $ f is many-one

$\Rightarrow $ f has no inverse.

(ii) Here f (5) = f (7) = 4

$\therefore $ f is not one-one [$\therefore $5 and 7 have same image 4]

$\Rightarrow $ f is many-one

$\Rightarrow $ f has no inverse.

(iii) Here each element of {2, 3, 4, 5} has a unique element
in {7, 9, 11, 13}.

Similarly each element of {7, 9, 11, 13} has a unique preimage
in {2, 3, 4, 5}.

$\Rightarrow $h is one-one onto $\Rightarrow $ f is invertible.

Hence, ‘h’ has the inverse.

##### Question 46:

Show that : f [– 1, 1] $\to $ R given by :

##### Answer:

f (x_{1}) = f (x_{2} )

##### Question 47:

Consider f : R $\to $ R given by f (x) = 4x + 3. Show that f
is invertible.

Find the inverse of f.

##### Answer:

We have : f (x) = 4x + 3.

##### Question 48:

Let f : R $\to $ R be defined by f (x) = 3x – 7. Show that f
is invertible.

Find f ^{–1} : R $\to $ R.

##### Answer:

We have : f (x) = 3x – 7.

Now f (x_{1}) = f (x_{2} )

$\Rightarrow $ 3x_{1} – 7 = 3x_{2} – 7

$\Rightarrow $ x_{1} = x_{2} $\Rightarrow $ f is one-one

$\Rightarrow $ f is invertible.

Let y = f (x) = 3x – 7

##### Question 49:

Consider R $\to $ [4, $\infty $) given by f (x) = x^{2} + 4. Show
that f is invertible with the inverse –1 f of f given by f ^{–1} (y) =

where R_{*} is the set of all nonnegative

real numbers.

##### Answer:

f (x_{1})= f (x_{2})

##### Question 50:

Consider f : R $\to $ [– 5, $\infty $), given by :

f (x) = 9x^{2} + 6x – 5.

##### Answer:

f (x_{1}) = f (x_{2} )

##### Question 51:

Let f : X $\to $ Y be an invertible function. Show that f has unique inverse.

##### Answer:

Since f is invertible, [Given]

$\therefore $ f is one-one onto.

If ‘g’ be the inverse of ‘f’, then

gof (x) = I_{X} and fog (y) = I_{Y}.

Let ‘g_{1}’, and ‘g_{2}’ be two inverses of ‘f’.

$\therefore $ fog_{1} (y) = I_{Y} and fog2 (y) = I_{Y}

$\Rightarrow $ g_{1} (y) = g_{2} (y)

[Q f is one-one onto]

Hence ‘f’ has a unique inverse.

##### Question 52:

Consider f : {1, 2, 3} $\to $ {a, b, c}, given by :

f (1) = a, f (2) = b and f (3) = c.

Find f ^{–1} and show that (f ^{–1} )^{–1} = f.

##### Answer:

We have : f (1) = a, f (2) = b, f (3)= c.

Thus f = {(1, a), (2, b), (3, c)}.

Clearly, f is one-one $\Rightarrow $ f is invertible

and 1 = f^{−1} (a), 2 = f^{−1} (b),

3 = f^{−1} (c)

$\Rightarrow $ f^{−1} : {a, b, c} $\to $ {1, 2, 3}.

Clearly, f^{−1} is one-one and onto

and ( f^{−1})^{−1} ={(1, a), (2, b), (3, c)}

[$\therefore $ f^{−1} = {(a, 1), (b, 2), (c, 3)}]

$\Rightarrow $ ( f ^{−1})^{−1} = f.

##### Question 53:

Let f : X $\to $ Y be an invertible function. Show that the
inverse of f^{–1} is f i.e. (f^{–1} )^{–1} = f.

##### Answer:

f : X $\to $ Y is invertible

$\Rightarrow $ f is one-one and onto.

$\Rightarrow $ g = x.

Hence ( f_{−1})_{−1} = f.

##### Question 54:

If f : R $\to $ R be given by f (x) = (3 – x_{3} )_{1/3} , then f of (x) is :

- x
^{1/3} - x
^{3} - x
- (3 – x
^{3})

##### Answer:

x

##### Question 55:

##### Answer:

##### Question 56:

Let R be the relation in the set {1, 2, 3, 4}, given by :

R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}.

Then :

- R is reflexive and symmetric but not transitive
- R is reflexive and transitive but not symmetric
- R is symmetric and transitive but not reflexive
- R is an equivalence relation.

##### Answer:

R is reflexive and transitive but not symmetric

##### Question 57:

Let R be the relation in the set N given by :

R = {(a, b) : a = b – 2, b > 6}.

Then :

- (2, 4) $\in $ R
- (3, 8) $\in $ R
- (6, 8) $\in $ R
- (8, 7) $\in $ R.

##### Answer:

(6, 8) $\in $ R

##### Question 58:

Let A = {1, 2, 3}. Then number of relations containing {1, 2} and {1, 3}, which are reflexive and symmetric but not transitive is :

- 1
- 2
- 3
- 4

##### Answer:

1

##### Question 59:

Let A = {1, 2, 3}. Then the number of equivalence relations containing (1, 2) is :

- 1
- 2
- 3
- 4

##### Answer:

2

##### Question 60:

Let f : R $\to $ R be defined as f (x) = x^{4}. Then :

- f is one–one onto
- f is many–one onto
- f is one–one but not onto
- f is neither one–one nor onto.

##### Answer:

f is neither one–one nor onto.

##### Question 61:

Let f : R $\to $ R be defined as f (x) = 3x. Then :

- f is one–one onto
- f is many–one onto
- f is one–one but not onto
- f is neither one–one nor onto.

##### Answer:

f is one–one onto

##### Question 62:

If f : R $\to $ R be given by f (x) = (3 − x^{31/3}, then fof (x) is :

- x
^{1/3} - x
^{3} - x
- 3 – x
^{3}

##### Answer:

x

##### Question 63:

##### Answer:

##### Question 64:

Consider a binary operation ‘*’ on N defined as :

a * b = a^{3} + b^{3}. Then :

- is ‘*’ both associative and commutative ?
- is ‘*’ commutative but not associative ?
- is ‘*’ associative but not commutative ?
- Is ‘*’ neither commutative nor associative ?

##### Answer:

is ‘*’ commutative but not associative ?

##### Question 65:

Number of binary operations on the set {a, b} is :

- 10
- 16
- 20
- 8

##### Answer:

16