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)

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

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**Aquameatwad** 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.

Read the question carefully.

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.

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

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

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

$\displaystyle 5671=5\cdot 10^3+6\cdot 10^2+{\color{blue}7}\cdot 10^1+1\cdot 10^0.$

Thus $\displaystyle 7\cdot 10$ tells that $\displaystyle 7$ is *the *__tens__ digit in $\displaystyle 5671$.

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?

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

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**Aquameatwad** 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.

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.?

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

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**Aquameatwad** 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 $\displaystyle d\in\{0,1,2,3,4,5,6,7,8,9\}$ then $\displaystyle d/\mathcal{R}$ is the set of integers having $\displaystyle d$ as its tens digit.

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

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?

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

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

The function $\displaystyle 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*.

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

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**Aquameatwad** i started by showing R={(10,10),(10,11),(10,12)....(20,20),(20,21)....( xn,yn),(yn,xn)}

It's hard to give an exhaustive picture with that notation unless you treat integers like a sequence of digits. Using a formula is better.

Also, an equivalence relation can be reframed as a partition, like: R/= : { {0, 1, -2, 100, -101, 506, -1302... }, {10, 18, 419, 1517, -617...}, ... } where there are ten sets, one for each group of numbers that are "equivalent" under the relation.

To show a), you need to show the relation is reflexive, symmetric, and transitive. Show each one and you have it.

b) wants you to describe all of the numbers that are in the same equivalence class as the number 3.

c) yields ten partitions, one for each class. Again, these are easy to describe informally, or by formula.