Determine which of the following define equivalence relation

Determine which of the following define equivalence relations in R²
If the relation fails to be an equivalence relation, identify at least one property that does not hold and give an example to demonstrate. If the relation is indeed an equivalence relation, give a geometrical interpretation of the quotient set.
(a.) (a,b)(c,d) if and only if a+2b=c-2d.

(b.) (a,b) (c,d) if and only if 2ab=2cd

Solution

(a)

1+2*(1) != 1-2*(1)

=>
(1,1) is NOT related to (1,1)

=>

R is NOT reflexive, hence NOT equivalent

(b)

2*a*b = 2*a*b

=>

(a,b) ~(a,b) => R is reflexive....(1)

2*a*b = 2*b*a

=>

(a,b) = (b,a) => R is symmetric .....(2)

let (a,b) ~(c,d), (c,d)~(e,f)

=>

2ab = 2cd

2cd = 2ef

=>

2ab = 2ef

=>

(a,b) ~(e,f)

=>

R is transitive ...(3)

(1),(2), (3) implies

R is Equivalence relation

geometric intepretation :

a hyperbola (since for (x,y) 2xy=k is an equation for a hyperbola)