Define a relation on Z by x y iff 2xy Prove that is an eq

Define a relation ~ on Z by x ~ y iff 2|(x+y).

Prove that ~ is an equivalence relation.

I know I have to prove the three properties (reflexivity, symmetry, and transivity) but I\'m really lost on how to even go about this or go through it all together. My professor was not good at explaining it. Please help!

Solution

~ is a relation defined on the set of integers as

x ~ y iff 2|(x+y).

Consider x and x

x+x = 2x is divisible by 2. So x ~x

So ~ is reflexive

-----------------------------

ii) x~y implies 2\\x+y which in turn implies 2\\y+x

So y~x

Hence symmetric

-----------------------------------------

iii) x~y and y ~ z implies

2\\x+y and 2 \\y+z

Adding 2\\x+2y+z which implies

2\\x+z (since y is an integer )

So transitive.

Since ~ is reflexive, symmetric and transitive ~ is an equivalence relation in Z