00:01
So in this problem in part a, we have the relation r, which is defined on z, the set of integers, such that x comma y is in the relation r if and only if 3 divides x plus 2y.
00:18
Now we'll show first that r is reflexive.
00:24
Well, ultimately we need to show that r is an equivalence relation.
00:28
So to show that, we'll show that r is reflexive, symmetric and transitive.
00:33
So first we deal with reflexive.
00:36
So note that for any integer a, a plus 2a is 3 times a, which is of course divisible by 3.
00:45
So we have that a comma a is in the relation r for all integers a.
00:55
So r is indeed reflexive.
01:02
Now we'll show that r is symmetric.
01:07
Suppose a comma b is in the relation.
01:11
R.
01:12
That means that 3 divides a plus 2b from the definition of r.
01:19
That is a plus 2b, we can write that as 3 times k for some integer k.
01:25
Now note that b plus 2a equals 3a plus 3b minus a plus 2b and we can write that as 3a plus b minus 3 times k because a plus 2b is 3 times k here and that is equal to three times a plus b minus k and that is of course divisible by three so we have shown that whenever a plus 2b is divisible by 3 so is b plus 2a that is whenever a comma b is in the relation r we have that b comma a is also in the relation r that is r is indeed symmetric now next will show that r is transitive.
02:15
To show that, first we assume that a comma b is in r and b comma c is also in r.
02:22
So the first implies that a plus 2b is divisible by 3, so we can write that as 3 times k for some integer k, and b plus 2c is also divisible by 3.
02:36
So we can write that as 3 times l for some other integer l, say.
02:41
And adding these two equations together, we have a plus 2b plus b plus b plus 2c equals 3 times k plus l.
02:51
Now here 2b plus b equals 3b.
02:54
So we have a plus 2c equals 3 times k plus l minus b.
03:01
So we subtract both sides by minus 3b.
03:04
So we have this.
03:05
So this shows that a plus 2c is divisible by 3.
03:11
That is a comma c is in the relation r.
03:15
So we have shown that whenever a comma b is in r and b comma c is in r, then it must also be true that a comma c is in r and that shows that r is indeed transitive.
03:30
So we have shown r is reflexive, symmetric and transitive.
03:34
So from all the above we can conclude that r is indeed an equal equivalence relation.
03:45
Now we are going to find the equivalence classes of r.
03:49
So note that a plus 2b is divisible by 3, that is a plus 2b equals 3k, if and only if a minus b equals 3k minus b.
04:03
So what we are doing is that subtracting both sides of this equation by minus 3b, and then a minus b equals 3 times k minus b, that is a plus 2b is divisible by 3 if and only if a minus b is also divisible by 3 and what that means is that a comma b is in the relation are if and only if three divides a minus b that is a and b leave the same remender when divided by 3 so we have that the equivalence classes are simply the numbers which leave the same remender by 3 when divided by 3 so we have an equivalence class for each remainder of three.
04:55
So anything divided by three can have remender 0, 1 or 2.
05:00
So we have equivalence classes corresponding to those.
05:05
So that is, z can be partitioned.
05:09
The set of integers can be partitioned into the distinct equivalence classes of r, which are 3k, such that k is in z.
05:20
So these are the numbers which are divisible by 3.
05:23
Integers and then we have 3k plus 1 where k is any integer so these are the numbers then the integers which leave the remainder 1 when divided by 3 and next we have 3k plus 2 where k is an integer and these are all the numbers which leave the remainder 2 when divided by 3 and that's it for part a now in part b here we have a different relation r, which is defined on the set of natural numbers, that is the set of positive integers, and it's given by x comma y is in the relation r if and only if x squared plus y squared is even.
06:13
So we are going to show that this r is also an equivalence relation, and to show that we'll show that r is a reflexive, symmetric, and transitive...