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

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S seema garhwal
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.

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S seema garhwal
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...

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S seema garhwal
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...

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S seema garhwal
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...

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S seema garhwal

$R = \{(P, Q) : \;distance \;of \;the\; point\; P\; from \;the \;origin \;is \;same \;as \;the\; distance \;of \;the \;point \;Q \;from \;the \;origin\}$

The distance of point P from origin is always same as distance of same point P from origin i.e.$(P,P)\in R$

$\therefore$ R is reflexive.

Let $(P,Q)\in R$ 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. $(Q,R)\in R$

$\therefore$R is symmetric.

Let    $(P,Q)\in R$     and    $(Q,S)\in R$

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.  $(P,S)\in R$

$\therefore$  R is transitive.

Hence, R  is an equivalence relation.

The set of all points related to a point $P \neq (0, 0)$ 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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S seema garhwal
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.

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S seema garhwal
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.

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S seema garhwal
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.

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S seema garhwal
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.

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

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S seema garhwal
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}

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S seema garhwal
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....

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S seema garhwal
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...

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S seema garhwal
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...

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S seema garhwal
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.

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S seema garhwal
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.

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S seema garhwal
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.

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S seema garhwal
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.

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S seema garhwal
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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