Question

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint: Look at the ratio a/b corresponding to (a, b).

Answer 1

Given that,

R is the relation on​​​​​ Z⁺ * Z⁺ . _(a,b) R_(c,d)<=> ad=bc

(a)To prove:

R is equivalence relation

proof:

For equivalence relation we have to prove reflexive, symmentric and transistive.

Reflexive:

Consider (a,b)R . _(a,b) R_{(a,b) is reflexive because ab=ab.}

Symmentric:

Consider (a,b) and (c,d) R . Given that ad=bc and obviously bc=ad. Therefore, R is symmentric i.e., _(a,b) R_{(c,d) =(c,d)} R _{(a,b) .}

Transistive:

Consider (a,b) ,(c,d) and (e,f) R.

_(a,b)R_(e,f) is af=be=>a/b=e/f ----(1)

_(e,f)R_{(c,d) is ed=fc => e/f=c/d -------(2)}

From (1) and (2)

a/b=c/d

ad=bc =>_(a,b)R_(c,d)

Therefore ,R is transistive.

R is an equivalence relation.

(b)  The equivalence class of (1,2)in R is { (2,4),(3,6),(4,8),(5,10),(6,13)}

Because 1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 6/12

(c) Equivalence class is the quotient set created from the relation divided by ~ .

R/~= Z⁺* Z⁺ /~

a/b~c/d is the equivalence class of R

It is also written as

R is the equivalence class of element (a,b)

{x R|x~ (a,b)}

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