# Thread: In Z define the relation xRy iff x and y have the same tens digit

1. ## In Z define the relation xRy iff x and y have the same tens digit

In Z define the relation xRy iff x and y have the same tens digit

a.)Show that R is an equivalence relation on Z
b.)Describe the set 3/R
c.)Describe the partition induced by the equivalence relation.

This was a question i had on a test which i got wrong

i started by showing R={(10,10),(10,11),(10,12)....(20,20),(20,21)....( xn,yn),(yn,xn)}
which im not sure is correct. and b and c i just got plain wrong.

Knowing the answer is just to my benefit. I spent a good amount of time on this.

2. ## Re: In Z define the relation xRy iff x and y have the same tens digit

In Z define the relation xRy iff x and y have the same tens digit
a.)Show that R is an equivalence relation on Z
b.)Describe the set 3/R
c.)Describe the partition induced by the equivalence relation.
The integer 5671 has the tens digit of 7 so it is related to 72.

a) It is easy to show a relation containing "same as" is an equivalence.

b) The number 3 has a tens digit of 0.

c) There are only ten equivalence classes.

3. ## Re: In Z define the relation xRy iff x and y have the same tens digit

can you elaborate more? isnt 1 in 5671 the ones place? i guess im not understanding the initial question

4. ## Re: In Z define the relation xRy iff x and y have the same tens digit

can you elaborate more? isnt 1 in 5671 the ones place? i guess im not understanding the initial question
$5671=5\cdot 10^3+6\cdot 10^2+{\color{blue}7}\cdot 10^1+1\cdot 10^0.$
Thus $7\cdot 10$ tells that $7$ is the tens digit in $5671$.

5. ## Re: In Z define the relation xRy iff x and y have the same tens digit

so refering to b. since the tens digit is 0 then 3/R is the partition of all numbers wih 0 in the tens digit? And c. the partition are 0-9 which are 10 partitions total?

6. ## Re: In Z define the relation xRy iff x and y have the same tens digit

so refering to b. since the tens digit is 0 then 3/R is the partition of all numbers wih 0 in the tens digit? And c. the partition are 0-9 which are 10 partitions total?
That is correct the way I read the question.

7. ## Re: In Z define the relation xRy iff x and y have the same tens digit

just curious, but what would be a mathematical way to write the answer to b and c.?

8. ## Re: In Z define the relation xRy iff x and y have the same tens digit

just curious, but what would be a mathematical way to write the answer to b and c.?
I don't know exactly what you mean by a mathematical way.
You could say, if $d\in\{0,1,2,3,4,5,6,7,8,9\}$ then $d/\mathcal{R}$ is the set of integers having $d$ as its tens digit.

9. ## Re: In Z define the relation xRy iff x and y have the same tens digit

yes that way.

10. ## Re: In Z define the relation xRy iff x and y have the same tens digit

in the same way. how would you define the relation to begin with?

11. ## Re: In Z define the relation xRy iff x and y have the same tens digit

in the same way. how would you define the relation to begin with?
The function $T(n) = \left\lfloor {\frac{{\left| n \right| - 100\left\lfloor {\frac{{\left| n\right|}}{{100}}} \right\rfloor }}{{10}}} \right\rfloor$ gives the tens digit of the integer n.