1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proving transitivity in equivalence relation a ~ b iff 2a+3b is div by 5

  1. Sep 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Relation on set of integers.

    a~b if and only if 2a+3b is divisible by 5
    show that ~ is an equivalence relation

    2. Relevant equations

    3. The attempt at a solution

    I have already proved that the relation is reflexive and symmetric, but I'm unsure of my approach at proving transitivity.

    if the relation is transitive, then: m~n and n~q implies m~q

    by the relation: 5 divides 2m + 3n, so let 2m+3n=5r
    5 divides 2n + 3q, so let 2n + 3q= 5t (r,t are integers)

    ==> 2m=5r-3n and 3q=5t-2n

    if m~q, then 2m+3q must be divisible by 5.
    2m + 3q= (5r-3n)+(5t-2n)= 5(r+t)-3n-2n
    = 5(r+t)-5n

    sum of numbers divisible by "a" = a number divisible by "a", so 5(r+t)-5n is divisible by 5, implying that 2m+3q is divisible by 5 which implies that m~q, which proves transitivity of the relation.


    Something tells me this proof is off. Please help me out
  2. jcsd
  3. Sep 12, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Nothing wrong with that argument.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook