Equivalence Relation

Question

Consider the following relations:

$R_1: ((a, b), (c, d)) belongs to R $ iff a + d = b + c

$R_2: ((a, b), (c, d)) belongs R$ iff ad = bc

Which of the following is equivalence relation.

Comments

@Magma007

@prateek Raghuvanshi

---

it's @Magma

let x = (a,b) , y = (b,c) , z = (c,d)

reflexive ,

xRx = (a,b) R (a,b) = a+b = b+a  [always true therefore , reflexive ]

symmetric :

xRy = (a,b) R (b,c) = a+c = b+b --> (i)

yRx = (b,c) R (a,b) = b+b = c+a ---> (ii)

i) and ii) are equal therefore , R is symmetric

Transitivity : 

xRy = a+c=b+b ----> i)

 yRz = (b,c)R(c,d) implies b+d=2c ----> 2

 xRz=(a,b)R(c,d) implies a+d=b+c --> 3

Now Adding 1 and 2 we get: a+c+b+d=2b+2c implies a+d=b+c which is to be proved for transitivity.

---

Your explanation is good enough.

But I generally do such questions by putting some random values.

Is that correct way to reach the answer as well?

---

I generally do such questions by putting some random values.

yeah I also do the same way 

---

@Magma @`JEET The value of x in the example should be ((a,b),(c,d)) as $R_{1}$ is $R\times R\rightarrow R\times R$ not $R\rightarrow R$.

Doesn't affect the answer but still !