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.
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.