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

• Dec 4th 2011, 09:09 PM
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. (Headbang)(Punch)(Crying)
• Dec 5th 2011, 02:36 AM
Plato
Re: In Z define the relation xRy iff x and y have the same tens digit
Quote:

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.
• Dec 5th 2011, 07:31 AM
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
• Dec 5th 2011, 07:39 AM
Plato
Re: In Z define the relation xRy iff x and y have the same tens digit
Quote:

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 $5671$.
• Dec 5th 2011, 08:52 AM
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?
• Dec 5th 2011, 09:01 AM
Plato
Re: In Z define the relation xRy iff x and y have the same tens digit
Quote:

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.
• Dec 5th 2011, 09:16 AM
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.?
• Dec 5th 2011, 09:27 AM
Plato
Re: In Z define the relation xRy iff x and y have the same tens digit
Quote:

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.
• Dec 5th 2011, 11:31 AM
Re: In Z define the relation xRy iff x and y have the same tens digit
yes that way.
• Dec 5th 2011, 05:44 PM
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?
• Dec 5th 2011, 06:05 PM
Plato
Re: In Z define the relation xRy iff x and y have the same tens digit
Quote:

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.
• Dec 5th 2011, 07:30 PM
Annatala
Re: In Z define the relation xRy iff x and y have the same tens digit
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