Question

Let A=NxN and define a relation R on A by (a,b)R(c,d) iff ab=cd ...show that R is an equivalence relation on A

Answer 1

so we need to show that the relation is:
symmetric
reflexive
transitive
did i spell right?

Answer 2

yup

Answer 3

fantastic

Answer 4

so we need to show
(a,b)R(a,b)
ab=ab
so the relation R is symmetric

Answer 5

I get confused with the (a,b)R(c,d) and it shows up in many questions,

Answer 6

(a,b)R(b,a)
ab=ba
so the relation R is reflexive

Answer 7

zarkon did i do something wrong

Answer 8

reflexive is (a,b)R(a,b)

Answer 9

ok maybe i got my properties mixed up alittle
i need to see

Answer 10

symmetric is (a,b)R(c,d) then (c,d)R(a,b)

Answer 11

oh ok, I think I get it now, the ordered pairs threw me off, I'm used to aRb and xRy and the like

Answer 12

darn it zarkon is right

Answer 13

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A:
a ~ a. (Reflexivity)
if a ~ b then b ~ a. (Symmetry)
if a ~ b and b ~ c then a ~ c. (Transitivity)

Answer 14

let me know if you need anymore help
i can help with this one now since i have the definition lol

Answer 15

ok, find an equivalence class for E(9,2)

Answer 16

I more so need to understand the concepts then the answers,

Answer 17

can you think of anything that is equivalent to (9,2)

Answer 18

so like (6,3), (18,1)?

Answer 19

yes

Answer 20

now just list them all

Answer 21

(1,18), (3,6), (2,9)...are there others?

Answer 22

(18,1), (6,3), (9,2)

Answer 23

how are they refliexive symmetric and transitive though? I know the dfinitions of them but how in this example are they those 3 things?

Answer 24

the relation R is those 3 things

Answer 25

ok, I understand how it's reflexive, but how is it symmetric and transitive? how can we say that (a,b)R(b,a)

Answer 26

"=" is reflexive symmetric and transitive

Answer 27

oh, is it because ab=cd is commutative?

Answer 28

sorry but could you just show me how R is transitive?

Answer 29

assume (a,b)R(c,d) and (c,d)R(e,f)

then ab=cd and cd=ef

thus ab=cd=ef hence ab=ef

so (a,b)R(e,f)

Answer 30

thanks

Answer 31

then I'm asked to find equivalence class with 2 elements

Answer 32

what is E(1,2)?