(A) Since, so
(B) Since, so
(C) Since, and so
(d) Since, so
The correct answer is option C

A =
For every there is .
R is reflexive.
Given, but
R is not symmetric.
For there are
R is transitive.
Hence, R is reflexive and transitive but not symmetric.
The correct answer is option B.

All lines are parallel to itself, so it is reflexive.
Let,
i.e.L1 is parallel to T2.
L1 is parallel to L2 is same as L2 is parallel to L1 i.e.
Hence,it is symmetric.
Let,
and i.e. L1 is parallel to L2 and L2 is parallel to L3 .
L1 is parallel to L3 i.e.
Hence, it is transitive,
Thus, , is equivalence relation.
The set of all lines related to the line are lines parallel...

Same polygon has same number of sides with itself,i.e. , so it is reflexive.
Let,
i.e.P1 have same number of sides as P2
P1 have same number of sides as P2 is same as P2 have same number of sides as P1 i.e.
Hence,it is symmetric.
Let,
and i.e. P1 have same number of sides as P2 and P2 have same number of sides as P3
P1 have same number of sides as P3 i.e.
Hence, it is...

All triangles are similar to itself, so it is reflexive.
Let,
i.e.T1 is similar to T2
T1 is similar to T2 is same asT2 is similar to T1 i.e.
Hence,it is symmetric.
Let,
and i.e. T1 is similar to T2 and T2 is similar toT3 .
T1 is similar toT3 i.e.
Hence, it is transitive,
Thus, , is equivalence relation.
Now , we see ratio of sides of triangle T1 andT3 are as shown
i.e. ratios...

The distance of point P from origin is always same as distance of same point P from origin i.e.

R is reflexive.

Let i.e. distance of the point P from the origin is same as the distance of the point Q from the origin.

this is same as : distance of the point Q from the origin is same as the distance of the point P from the origin i.e.

R is symmetric.

Let and

i.e. distance of the point P from the origin is same as the distance of the point Q from the origin, and aslo distance of the point Q from the origin is same as the distance of the point S from the origin.

We can say that distance of point P,Q,S from origin is same.Means distance of point P from origin is same as distance of point S from origin i.e.

R is transitive.

Hence, R is an equivalence relation.

The set of all points related to a point are points whose distance from origin is same as distance of point P from origin.

In other words we can say there be a point O(0,0) as origin and distance between point O and point P be k=OP then set of all points related to P is at distance k from origin.

Hence, these set of points form circle with centre as origin and this circle passes through point P.

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Let there be a relation A in R
So R is not reflexive.
We can see and
So it is symmetric.
Let and
Also
Hence, it is transitive.
Thus, it Symmetric and transitive but not reflexive.

Let there be a relation R in R
because
Let i.e.
But i.e.
So it is not symmetric.
Let i.e. and i.e.
This can be written as i.e. implies
Hence, it is transitive.
Thus, it is Reflexive and transitive but not symmetric.

Let
Define a relation R on A as
If , i.e.. So it is reflexive.
If , and i.e.. So it is symmetric.
and i.e. . and
But So it is not transitive.
Hence, it is Reflexive and symmetric but not transitive.

Let
Now for , so it is not reflexive.
Let i.e.
Then is not possible i.e. . So it is not symmetric.
Let i.e. and i.e.
we can write this as
Hence, i.e. . So it is transitive.
Hence, it is transitive but neither reflexive nor symmetric.

Let
so it is not reflexive.
and so it is symmetric.
but so it is not transitive.
Hence, symmetric but neither reflexive nor transitive.

For , as
Henec, it is reflexive.
Let, i.e.
i.e.
Hence, it is symmetric.
Let, i.e. and i.e.
i.e.
Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The set of all elements related to 1 is {1}

For , as which is multiple of 4.
Henec, it is reflexive.
Let, i.e. is multiple of 4.
then is also multiple of 4 because = i.e.
Hence, it is symmetric.
Let, i.e. is multiple of 4 and i.e. is multiple of 4 .
is multiple of 4 and is multiple of 4
is multiple of 4
is multiple of 4 i.e.
Hence, it is transitive.
Thus, it is reflexive,symmetric and transitive i.e....

Let there be then as which is even number. Hence, it is reflexive
Let where then as
Hence, it is symmetric
Now, let
are even number i.e. are even
then, is even (sum of even integer is even)
So, . Hence, it is transitive.
Thus, it is reflexive, symmetric and transitive i.e. it is an equivalence relation.
The elements of are related to each other because the...

A = all the books in a library of a college
because x and x have same number of pages so it is reflexive.
Let means x and y have same number of pages.
Since,y and x have same number of pages so .
Hence, it is symmetric.
Let means x and y have same number of pages.
and means y and z have same number of pages.
This states,x and z also have same number of pages i.e.
Hence, it is...

Q. 6 Show that the relation R in the set given by is

symmetric but neither reflexive nor transitive.

Let A=
We can see so it is not reflexive.
As so it is symmetric.
But so it is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.

because
So, it is not symmetric
Now, because
but because
It is not symmetric
as .
But, because
So it is not transitive
Thus, it is neither reflexive, nor symmetric,nortransitive.

As so it is reflexive.
Now we take an example
as
But because .
So,it is not symmetric.
Now if we take,
Than, because
So, it is transitive.
Hence, we can say that it is reflexive and transitive but not symmetric.

R defined in the set
Since, so it is not reflexive.
but
So,it is not symmetric
but
So,it is not transitive.
Hence, it is neither reflexive, nor symmetric, nor transitive.

Taking
and
So,R is not reflexive.
Now,
because .
But, i.e. 4 is not less than 1
So,
Hence, it is not symmetric.
as
Since because
Hence, it is not transitive.
Thus, we can conclude that it is neither reflexive, nor symmetric ,nor transitive.

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