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Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p,q),(r,s)) \in R$ if and only if $p-s=q-r$. Which one of the following is true about $R$?

  1. Both reflexive and symmetric
  2. Reflexive but not symmetric
  3. Not reflexive but symmetric
  4. Neither reflexive nor symmetric
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7 Answers

Best answer
89 votes
89 votes

The key trick here is to realize that the relation is of the form :

 {ordered pair, ordered pair} and not simply ordered pair.

Ok, so for reflexive

$\forall_{a,b}\, if((a,b),(a,b)) \in \mathrel{R} \rightarrow \text{reflexive}$

$((a,b),(a,b)) \in \mathrel{R} \;\leftrightarrow(a-b=b-a) $ (not possible for any postive integers b and a) 

But that is a contradiction hence it is not reflexive.

Now, for symmetric

$((a,b),(c,d))\in \mathrel{R}\rightarrow((c,d),(a,b))\in \mathrel{R}$

$((a,b),(c,d))\in\mathrel{R} \rightarrow(a-d=b-c)$

$((c,d),(a,b))\in\mathrel{R}$

$\because (c-b=d-a)\leftrightarrow (d-a=c-b)\leftrightarrow(-(a-d)=-(b-c))\leftrightarrow(a-d=b-c)$

So, it is symmetric.

Hence, C is the correct option.

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19 votes
19 votes
  1. take example for symmetric relation ... for p=6, q=4,r=4, s=6, (6,4)(4,6) 

p-s=6-6=0
q-r=4-4=0
so if we will check the condition (means p-s and q-r) for symmetric relation then p-s =q-r will b same always 

     
now we can see the relation (6,4)(4,6) is not reflexive but exist  in the relation R. so we can say that R may or may not b reflexive.
so option C is correct Not reflective but symmetric

 

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8 votes
8 votes
Let us first rearrange the equation to make it look simple

p−s=q−r

p+r=q+s // which means sum of first elements is same as sum of last two elements.

Reflexive:-

(p,q)R(p,q) iff p+p=q+q ,which is not true always so not reflexive

Symmetric.

If (p,q)R(r,s) => (r,s)R(p,q)

p+r=q+s       =>  r+p=s+q

which will always hold.Hence symmetric

So, c is correct
2 votes
2 votes
((p, q), (r, s)) ∈ R if and only if p–s = q–r

(p, q) is not related to (p, q)

as p-q is not same as q-p.

The relation is symmetric because if p–s = q–r, then s-q = s-p. 
Answer:

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